Think of potential as like potential energy. If the mobile charges are electrons and they are mostly at rest they will mostly move towards lower potential energy which is higher potential.
So think of it as like a hill with electrons free to roll down hill (higher potential) until they get to a surface of the conductor at which point they are not free to move. So, you can imagine a fence built on the surface of the conductor that keeps things from leaving. And you can imagine some hills inside.
If the hills inside were flat, then indeed you could have some electrons at the edge and everything could sit there in statics. And if the hill inside was actually flat, then things can stay there.
There is a mountain called Mount Saint Helens. It looks like a normal mountain except the top looks like it was just chopped off (it actually exploded in a huge eruption in 1980).
The edge of the top is all at the same height; things on the edge would fall (and move away) if they were free to move. This is exactly what is going on in a conductor.
Now imagine you poured water on the top. The water is free to move if the pile of water gets higher than the region around. Real water has some surface tension but the point is that it can even itself come out because a region of higher water is free to move.
The same thing happens in the conductor. If you had an isolated positive charge, it would create a positive potential about itself which is a negative potential energy around itself. This would attract electrons.
So if electrons moved away from a positive charge leaving a positive charge all by itself, that would not be creating equilibrium because it would attract more electrons to it.
Charges can move. If a charge moved from the surface to another point inside and the charge there moved to a other point inside and so for until eventually a charge near another part of the surface moves to the surface, that is all fine.
So do that. Draw a bunch of surfaces of constant potentiality in red. Then draw a bunch of curves in red that are ways orthogonal to the surfaces, these are the field lines of the electric field. Now there is no force pushing in the direction of the equipotential and the force is all in the direction of the field.
Just because there is a force in a direction doesn't mean that charges go that way; that depends on the initial velocity of the charge. But imagine a region large enough to have many charges in it, the average velocity of the mobile charges could be zero because they bump into the non mobile charges except in the direction of the electric field. There they could have a non-zero average because they give and get from the non mobile charges but they consistently gain momentum in the direction of the electric field.
So now imagine a current everywhere pointing in the direction of the electric field. If you are in electrostatics then the field lines don't make any loops. So they can start at one part of the surface and end on another part of the surface. That is exactly how charge can flow.
This is important. This is saying that if all the charges were placed on the surface in a way that wasn't an equipotential and all the charges were kept at rest and were kept there for some reason so they are there in place long enough for say the speed of light to traverse across the whole object then there would then be electrostatic fields throughout the object.
If you now let go and let all the mobile charges move inside the conductor then the field at that moment is electrostatic and pointed so that current will flow through the body so that charge on the surface flows to other parts of the surface. And it never stops anywhere inside the object.
This is so frustrating. I've basically described how if you add charge to a conductor that has no net charge on the inside then the charge moves around in such a way that it continues to have no net charge on the inside.
I'm saying that your concern about the positive charge never happens in the first place. How rude. But there is some truth. If conductors start with the property of having no net charge in the body of the conductor then they can continue to have that property. That is essential for electrostatics and all statics in general.
I talked about their being no loops that really just means we can keep magnetism out of it (loops of electric fields is associated with changing magnetic fields). But we really had the electric field line go from one part of the surface to the other because there was no non-zero charge density to terminate on.
So what if for some reason you just made a conductor or just started getting to statics. Then there might be charge in the body. Now if you still have the current be proportional to the electric field you get for instance $\vec J=\sigma \vec E$ then you can take the divergence of both sides and get $$\sigma\frac{\rho}{\epsilon_0}=\sigma\vec \nabla \cdot \vec E = \vec \nabla \cdot \vec J$$
Where we used the Maxwell equation $\dfrac{\rho}{\epsilon_0}=\vec \nabla \cdot \vec E$ and we can also take the divergence of $$\vec \nabla \times \vec B=\mu_0\vec J+\mu_0\epsilon_0\frac{\partial \vec E}{\partial t}$$ to get the continuity equation
$$\vec \nabla \cdot \vec J=-\epsilon_0\vec \nabla \cdot\frac{\partial \vec E}{\partial t}=-\frac{\partial \rho}{\partial t}.$$
This means we have $$\frac{\partial \rho}{\partial t}=-\vec \nabla \cdot \vec J=-\frac{\sigma}{\epsilon_0}\rho.$$
Now we have stepped out of electrostatics but that is because we are talking about how it got that way. But now we see the charge density decreases when it is positive and increases when it is negative and in fact it approaches zero exponentially. So a conductor is never perfect you can think of it as something with a super huge $\sigma$ and so it gets super close to no charge in the body in almost no time.
Now, there is another reason I didn't go straight to this. There is no law of physics that says $\vec J=\sigma \vec E$ and in fact even materials that behave approximately like that have the value of $\sigma$ depend on the frequency of the change if it changes or even just depending on the temperature (and current changes the temperature).
If you placed a huge amount of charge deep inside a conductor by say shooting electrons going at almost the speed of light at it. Then yes the conductor might heat and buckle and strain and generally take some time to get that charge to the outside if it was a tremendous amount of charge. No real material is perfect.
But to the degree that the charges follow the electric field then when there is a charge density the field lines will be pointing the direction as to attract a countering charge.
Which is what we started with. If you had an imbalanced positive charge, it will attract mobile electrons to it and reduce the charge imbalance. If you have an excess of electrons in a spot they will push the neighbour mobile charges away from it and thus reduce the density in that region that contained them all.
And the continuous charge density is always for a region that contains many charges it isn't a thing that jumps up at every proton and down at every electron.
So it approaches zero by attracting more mobile charges into it or pushing more mobile charges out of it depending on whether the field diverges there in a positive or negative way and it does that based on whether there is a positive or negative charge density there.
What did Feynman want to tell in the bold lines?
If the equipotential surface wasn't everywhere tangent to the surface then it would dip inside. Sort of like if you divided your room in two and told your sibling not to cross to your half (TV sitcom style) and your brother or sister blew a bubble. If the bubble was tangent to the imaginary surface it wasn't supposed to cross; but if it isn't tangent to that surface then you can follow the surface into your side.
So the equipotential would cross inside the conductor. Which means you can follow it inside to the body of the conductor and from there you can look at the normal to the equipotential and you'll see a field line going there. That field line either terminates on a charge density (where the electric field lines terminate) or goes to the surface of the conductor. So it has two ends either both in the surface, both on densities inside or from a density inside to the surface.
If current moves in the direction of electric fields then current can flow between the two ends. If they are both on the surface then one part of the surface will have less charge at the spot that field line hit it and the the other place the field line hit the surface gains it.
If one of the ends is inside then that charge density will approach the zero exponentially fast and the current will be causing it so an opposite charge will head to the other end of the field line. So two opposite charge densities inside could be weakening each other or a charge imbalance inside can be sending its excess to the surface. And both. A place where field lines terminate can have field lines from multiple directions coming out of it and they could end on different places.
It might help if you drew the fields for two oppositely charged point charges then draw an arbitrarily surface around then so it contains them both. Where those field lines go to each other they are discharging each other. Where those field lines hit the surface you drew is where current is liking up charge on the surface. Eventually all the charge is on the surface. That's a good example to look at.
But the example in the last paragraph doesn't explain what happens when you place charge on the surface. So draw the two charges again. Draw the field lines again. And draw a new surface, but this time draw a new surface but make sure it passes through one of them and contains the other one inside. Again, the current can follow the field lines so the charge that starts out on the surface travels through the body towards the opposite charge and some travels through the conductor from that charge to where the field line terminates on the surface and some actually travels along the surface as a surface current where the surface is tangent to the field. The two concentrations of charge discharge exponentially and you end up with current spread out on the surface.
OK. Now draw a surface where it intersects both charges on its surface. Here is an example where all the charge is in the surface. It moves through the conductor and sometimes along the surface but here is the key. The current doesn't have a nonzero divergence inside the conductor so even though current flows through the body the body never acquires a nonzero charge density. So the charge stays in the surface always even though current flows through the conductor.
Make sure you can see that. Current can flow even if there is no net charge density. So when Feynman says the charge moves around the surface he doesn't mean that there is can't be current inside. But the charge imbalance stays on the outside if there was no change imbalance inside. Charge can flow inside as long as equal amounts flow in and out of every region in the body of the conductor then no imbalance will form.
And as we saw earlier if there originally was a charge imbalance inside then it decays away exponentially. This is how an ohmic material ($\vec J=\sigma \vec E$) behaves. A different material might behaves slightly differently but if it is a good conductor it will behave similarly and you can think of a perfect conductor as an ohmic material with $\sigma=\infty$ so the charge on the surface just flash gets to that final configuration that an ohmic material would go to.
And that's how to think of a perfect conductor. Just imagine a material with $\vec J =\sigma \vec E$ and find out the state it approaches after an infinite amount of time. Then imagine your object gets super close to that in a very short time. So basically imagine that $\vec J =\sigma \vec E$ for a huge $\sigma$ even if your material isn't as simple as having $\vec J =\sigma \vec E.$
Best Answer
I have a guess, although I don't know if it is correct.
If you model the globe as a simple insulating circular glass shell with a constant spatial charge density embedded in the glass in the immediate vicinity of the finger and solve Maxwell's equations numerically for the potential, you observe something like this:
The rationale for placing a nontrivial static charge density in the vicinity of the finger in contact with the spinning globe is simply because glass, the so-called "vitreous" static source, dislodges surface charges during mechanical abrasion, and the charges have nowhere to go, since glass is mostly nonconducting.
As you can see, on the inside and outside of the glass in the immediate vicinity of the finger, there is a potential gradient. As such, dielectric breakdown becomes a possibility. By Paschen's Law, dielectric breakdown is orders of magnitude more likely in the partially-evacuated interior of the globe; as such, surface charge in the vicinity of the finger on the interior of the globe are accelerated through the vacuum and redistribute themselves on the lower-potential walls of the globe farther away from the finger. The electron collisions with nitrogen and other rarefied molecules in the vicinity of this transient current generate the blue glow near the finger.
Since the globe is spinning and the finger continually moves across the globe, there is no need to worry about a net nonphysical change in the interior surface charge density during each traversal of the loop, since they are constantly being "spatially recycled" during the spinning cycle.
For this reason, I am willing to bet that a similar experiment conducted with a finger rubbing the same spot on an evacuated glass bulb will NOT produce a continual glow discharge in the vicinity of the finger, despite the mechanical similarity of the processes.
Finally, assuming my mechanism is correct, there should be no visible glow discharge if the glass bulb is near-perfectly evacuated. Vacuum techniques in Hauksbee's time were, of course, far from this degree of rarefaction.