I finally stumbled upon the term radar to describe this use of a reference timelike worldline. A good starting resource is: Perlick, Volker. "On the radar method in general-relativistic spacetimes.", which points to a bunch of other resources.
The $r^\star$ and $\tau^\star$ are in fact the radar distance and radar time resp.
I still haven't found any name for the surface, though radar bubble suggests itself. There is a region around the reference worldline where these bubbles are globally spacelike and topologically $S^2$ (a bubble). In flat space this region is the entire space, though a gravitating object will make bubbles self-intersect at certain distance.
A radar bubble can fail to be even locally spacelike: if a null geodesic intersects the worldline twice, then the corresponding bubble would include that geodesic. A black hole can do this. It can act as a gravitational mirror, and bounce light coming in on a certain angle from a distant source back.
Addendum
Instead of the light cones themselves, the above also works using the boundary of the chronological future (past). The chronological future (past) of an event $p$ are the events that can be reached from $p$ (can reach $p$) by a timelike path. They are designated $I^\pm(p)$, and their boundaries $\delta I^\pm(p)$.
In a globally hyperbolic spacetime, these are subsets of the past or future light cone, excluding the parts where it self-intersects.
The intersection of a future and past boundary (not to be confused with the boundary of the intersection) $\delta I^+(p) \cap \delta I^-(q)$, is indeed globally spacelike, though not always topologically $S^2$.
Given a timelike path $p(\lambda)$, let's label the bubble $B_p(\lambda,\mu) = \delta I^+(p(\lambda-\mu)) \cap \delta I^-(p(\lambda+\mu))$. Then for a given $\lambda$, the union of bubbles $\bigcup_{\mu \ge 0} B_p(\lambda,\mu)$ is a globally spacelike 3-D hypersurface. This family of hypersurfaces indexed by $\lambda$ foliates the part of the spacetime reachable from the path (unlike the "instantaneous simultaneous spaces" -- even in special relativity, two instantaneous simultaneous spaces at different events of an accelerated worldline will intersect each other).
In particular this is true if the path parameter is just the proper time along the path (i.e. $\tau^\star = \lambda$ and $r^\star = \mu c$).
The fact that the curve doesn't have zero path length is identical to the following 'proof' that $\pi=4$.
A detailed explanation can be found in this link, but the main idea is that the black line doesn't become a tangent line in the limit. This means the perimeter of the circle and jagged line aren't equal in the limit.
To why this implies it is maximal: a regular function has the property that in its maximum $f(x+\delta x)\leq f(x)$. Here $x$ maximizes $f$ and $\delta x$ is a small (or infinitessimal) quantity. For the proper time this argument is less obvious because it depends on the entire path. You can define it as a functional: an object which takes a function as input and outputs a scalar.
$$\Delta \tau[x^\mu]=\int d\lambda\sqrt{-\eta_{\mu\nu}\frac {d x^\mu}{d\lambda}(\lambda)\frac {d x^\nu}{d\lambda}(\lambda)}$$
Our argument can then be extended to
$$\cases{\Delta \tau[x^\mu+\delta x^\mu]<\Delta \tau[x^\mu] & $x^\mu$ is a maximum\\
\Delta \tau[x^\mu+\delta x^\mu]>\Delta \tau[x^\mu] & $x^\mu$ is a minimum}$$
Now $\delta x^\mu(\lambda)$ it not a constant anymore but a function. In our case it is the offset between our geodesic and the jagged approximation of the geodesic. Since the proper time is positive and $\Delta \tau[x^\mu+\delta x^\mu]=0$ we have that $x^\mu$ must be a maximum.
Note: forgive me if I made mistakes, has been a while since I did any GR.
Best Answer
Even in curved spacetime, you can perform a coordinate transformation at any location ("move to a freely falling frame") such that your metric is locally flat , and takes the form \begin{equation} ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2\end{equation}
If you consider a null trajectory where $ds^2 = 0$, then the above equation takes the form
\begin{equation} cdt = \sqrt{dx^2 + dy^2 + dz^2}. \end{equation}
This is the statement that "the speed of light times the differential time interval, as measured by an observer in a freely falling frame at the location in consideration, is equal to the differential physical distance traveled along the trajectory, measured by that same observer." From Einstein's equivalence principle, this is precisely the way that light must behave.