The reason is that the light is the fastest possible way of transmission of information. Yes, before measuring one-way speed of light we have to synchronize spatially separated clocks and to synchronize clocks we need to know velocity of the one way speed of light.
But, we can measure two-way speed of light with a single clock.
To use the sun as a source of synchronization is the same so as to use an equidistant from clocks source of light. But, the moment they will see the sun can is very “subjective” and depends on one-way speed of light from the sun towards them, which can be different.
Let’s this laboratory moves relatively to this source of light and one-way speed of light is different leftward and rightward. When the beam comes to these clocks, you still have to assume what was the one-way velocity so as to set hands of these clocks. You still don’t know this one-way velocity and are free to assign it certain value. If you will assume, that one-way velocity was c, these clocks will be Einstein – synchronized, and measured one-way velocity will be exactly c, even if this synchronization is “wrong”.
It is well-known, that Einstein – synchronization is a special case of broader Reichenbach – synchronization, which is also self–consistent. Reichenbach synchronization allows anisotropic one-way velocities until two-way speed of light is c. For example, if in one direction it will be infinitely large in the other direction it will be very close to c/2.
Lorentz Ether theory (which is empirically equivalent to SR assumes, that the Earth moves in “preferred” frame, or Ether. Yes, according to this theory, as you say “light is NOT constant since we on earth are moving though space. & since we are moving though space we should observe speed of light as speed of light +- our speed of travel in the opposite direction.“ In Lorentz theory one way velocities of light are the same only in the Ether, but we cannot measure it because of well – known reasons. What we can measure, is two-way velocity, which is isotropic indeed, see Michelson Morley experiment. But, this experiment, as well as any other is not able to say something about one-way speed of light.
It is often said that one-way speed of light is conventional. So, time dilation in certain sense is also conventional, because it depends on synchronization procedure. If each observer synchronizes clocks by Einstein, they will see dilation of each other clock.
Let’s there are two relatively moving reference frames, S and S’. Let’s S’ moves very close to c. If clocks in these frames synchronized by Einstein, they will be “slower” each other.
If in the frame S the observer synchronizes clocks by Einstein and the observer in S’ synchronizes clocks by Reichenbach, assuming that he himself is moving in the frame S and one- way speed of light is anisotropic in his frame (since it is isotropic in S), this observer S’ will see, that clock S is ticking $\gamma$ times faster.
There are a couple of free sources that compare “isotropic one-way ” relativity and “anisotropic one” :
On the simplest examples of floating in water ships this article or this book simulates all kinematic effects of Special Relativity, Lorentz transformations, anisotripic one-way speed of light and isotropic two-way, time dilation, length contraction, relativistic velocities addition, Relativistic Doppler effect, reciprocity of Lorentz transformations, Twin paradox, Bell’s spaceship paradox etc.
Another was: Janssen, Michel (1995), A Comparison between Lorentz's Ether Theory and Special Relativity in the Light of the Experiments of Trouton and Noble, but I was not able to find it right now.
Another well-known book: Max Jammer; Concepts of simultaneity: from Antiquity to Einstein and beyond - https://muse.jhu.edu/book/3280
Good to note, that the one way speed of light is anisotropic on rotating ring.
Does Sagnac effect imply anisotropy of speed of light in this inertial frame of reference?
This made me ask this question is taking the speed of light same in all directions an axiom of some sort?
Yes, although it is called a postulate rather than an axiom. This is Einstein's famous second postulate:
Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence
$${\rm velocity}=\frac{{\rm light\ path}}{{\rm time\ interval}} $$
where time interval is to be taken in the sense of the definition in § 1.
A. Einstein, 1905, "On the Electrodynamics of Moving Bodies"
https://www.fourmilab.ch/etexts/einstein/specrel/www/
This postulate is simply assumed to be true and the consequences are explored in his paper. The subsequent verification of many of the rather strange consequences is then taken to be strong empirical support justifying the postulate. This is the heart of the scientific method.
So are all of our physics theories based on the assumption and what would happen if light turns out to be moving at different speeds in different direction? Will that enable transfer of information faster than the speed of light and is there any way for us knowing that the transfer happens faster than the speed of light?
Yes, all of our physics theories are based on this assumption, but the assumption itself is simply a convention. The nice thing about conventions is that there is no "wrong" or "right" convention. This specific convention is known as the Einstein synchronization convention, and it is what the second postulate above referred to by "time interval is to be taken in the sense of the definition in § 1". From the same paper in section 1:
Let a ray of light start at the “A time” $t_{\rm A}$from A towards B, let it at the “B time” $t_{\rm B}$ be reflected at B in the direction of A, and arrive again at A at the “A time” $t'_{\rm A}$.
In accordance with definition the two clocks synchronize if $$t_{\rm B}-t_{\rm A}=t'_{\rm A}-t_{\rm B}$$
A. Einstein, 1905, "On the Electrodynamics of Moving Bodies" https://www.fourmilab.ch/etexts/einstein/specrel/www/
If we define $\Delta t_A= t'_A-t_A$ then with a little rearranging this becomes $t_B=\frac{1}{2}(t_A+t'_A)=t_A+\frac{1}{2}\Delta t_A$. This is a convention about what it means to synchronize two clocks. But it is not the only possible convention. In fact, Reichenbach extensively studied an alternative convention where $t_B=t_A+ \epsilon \Delta t_A$ where $0 \le \epsilon \le 1$. Einstein's convention is recovered for $\epsilon = \frac{1}{2}$ and the Veritasium video seemed oddly excited about $\epsilon = 1$.
Note that the choice of Reichenbach's $\epsilon$ directly determines the one way speed of light, without changing the two way speed of light. For Einstein's convention the one way speed of light is isotropic and equal to the two way speed of light, and for any other value the one way speed of light is anisotropic but in a very specific way that is sometimes called "conspiratorial anisotropy". It is anisotropic, but in a way that does not affect any physical measurement. Instead this synchronization convention causes other things like anisotropic time dilation and even anisotropic stress-free torsion which conspire to hide the anisotropic one way speed of light from having any experimental effects.
This is important because it implies two things. First, there is no way to determine by experiment the true value, there simply is no true value, this is not a fact of nature but a description of our coordinate system's synchronization convention, nature doesn't care about it. Second, you are free to select any value of $\epsilon$ and no experiment will contradict you.
This means that $\epsilon=\frac{1}{2}$ is a convention, just like the charge on an electron being negative is a convention and just like the right-hand rule is a convention. No physical prediction would change if we changed any of those conventions. However, in the case of $\epsilon=\frac{1}{2}$ a lot of calculations and formulas become very messy if you use a different convention. Since there is no point in making things unnecessarily messy, it is a pretty strong convention.
Finally, regarding FTL information transfer. If we use $\epsilon \ne \frac{1}{2}$ then there is some direction where information can travel faster than $c$. However, since in that direction light also travels faster than $c$ the information still does not travel faster than light. It is important to remember that under the $\epsilon \ne \frac{1}{2}$ convention the quantity $c$ is no longer the one way speed of light, so faster than light and faster than $c$ are no longer equivalent.
Best Answer
Yes, it is often assumed that Rømer measured the speed of light in one direction. It may seems trange, but Rømer velocity is also the velocity obtained under the tacit assumption of the equality of the speeds of light in opposite directions. The fact of the matter is that Rømer and Cassini were speculating about the movement of Jupiter’s satellites, automatically assuming that the observers’ space was isotropic.
The Estonian - Australian physicist Leo Karlov showed that Rømer actually measured the speed of light by implicitly making the assumption of the equality of the speeds of light back and forth.
L. Karlov, “Does Roemer's method yield a unidirectional speed of light?” Australian Journal of Physics 23, 243-258 (1970)
Also:
L. Karlov “Fact and Illusion in the speed of light determination of the Roemer type” American Journal of Physics, 49, 64-66 (1981)
Some reflections on the one-way speed of light are here.
Another interesting method to measure one - way speed of light that you may discover soon or later was so - called Double Fizeau Toothed wheel. That is two toothed wheel attached to opposite sides of long rotating shaft and a beam of light between the teeth. This method was employed (probably without proper due - diligence) by S. Marinov and M. D. Farid Ahmet.
However, Herbert Ives in his 1939 article "Theory of Double Fizeau toothed wheel" predicted that outcome of the measurement will be exactly c due to relativistic twist of the rotating shaft.