PeterDonis said:
We need to more carefully distinguish the actual postulates and assumptions here. And we need to bear in mind that the modern understanding of the postulates and assumptions has evolved, compared to the understanding in 1905 when Einstein published his SR paper.
In modern terms, the key postulates/assumptions that underlie SR are, first, that spacetime is a 4-dimensional manifold with one timelike and three spacelike dimensions, and second, that spacetime is flat. Both of those postulates/assumptions are independent of any choice of reference frame or coordinates (and they can be checked in any coordinate chart, there is no need to have an inertial frame), and they are also independent of any notion of "speed" or "invariant speed" (since the presence of null intervals and null cones in the geometry can be verified without any use of such notions).
The facts that we can construct inertial frames in this flat spacetime of SR that use the Einstein clock synchronization convention as their simultaneity convention, and that the metric has the same form in every such inertial frame, and that Lorentz transformations transform between these frames, and that the coordinate speed of any object with a null worldline is the same, ##c##, in all such frames, are all consequences of the above assumptions/postulates; they are not assumptions/postulates themselves. As consequences of the same assumptions/postulates, they do all go together, but no one of them is logically prior to any other of them. They are all just things that come out when you work out the consequences of the above assumptions/postulates.
So while it is true that, in any inertial frame, it will be the case that the Einstein clock synchronization convention provides the simultaneity convention, and that the coordinate speed of light is ##c## and is invariant under Lorentz transformations from one inertial frame to another, that doesn't mean the former is a logical consequence of the latter. It just means both are consequences of constructing an inertial frame in the flat Minkowski spacetime that satisfies the two key postulates/assumptions above.
Well, you can also start a bit more from the physics point of view and not postulate the spacetime geometry beforehand but derive it from symmetry considerations. I'd say that's the main common successful scheme of 20th-century physics starting with Einstein, put to solid mathematical foundations by Noether (based on general developments in 19th-century mathematics by Riemann, Klein, et al).
So you can start by just assuming the special principle of relativity and further that any inertial observer (i.e., any observer who has established an inertial frame of reference and is at rest wrt. this frame) finds a Euclidean spatial geometry and a homogeneous directed time. With the corresponding symmetries (translations in space and time, isotropy of space, symmetry under "boosts") you can derive that there are, up to isomorphy, two possible spacetime models: the Galilei-Newton spacetime (a fiber bundle) and Einstein-Minkowski spacetime (a pseudo-Euclidean 4D affine manifold with a fundamental form of signature (1,3) or (3,1) depending on your preference of conventions). The latter implies the existence of a universal "limiting speed". The rest is an empirical question, and to the best of our knowledge the electromagnetic field is a massless vector field (the current limit of the photon mass is ##m_{\gamma} < 10^{-18} \text{eV}## and thus this limiting speed is the phase velocity of free electromagnetic waves in vacuo.
Concerning the question about the Einstein synchronization condition using light clocks, I'm not sure, but I think it's pretty clear that it is one operational way to define "Galilean spacetime coordinates", i.e., the usual components of the spacetime four-vectors wrt. a Minkowski-orthonormal basis.
This is now all of course about special relativity. For the foundation of GR you have again two choices. One is Einstein's geometrical way, which after 10 years struggle boiled down to the assumption of a pseudo-Riemannian manifold with the fundamental form having the Lorentz signature (that's why for short such manifolds are also called Lorentz manifolds). The connection to the physics is Einstein's (strong) equivalence principle, leading to the Einstein field equations (in general with cosmological constant) quite uniquely (most elegantly derived by using Hilbert's argument about the action, see Landau Lifshitz vol. 2).
The other approach is more modern and considers GR as a gauge theory in analogy to the "gauging" of a global symmetry in relativistic QFT. The difference is that here what's "gauged", i.e., what's made local, is the Lorentz symmetry. Analyzing this idea together with the various tensor and spinor fields known special-relativistic field theory leads in the general case to a somewhat more general spacetime-structure, i.e., a socalled Einstein-Cartan manifold, i.e., a Lorentz manifold generalized to also have torsion. The connection is assumed to the compatible with the Lorentz-pseudometric but having torsion, which is dynamically determined as is the pseudometric. For our usual "macroscopic use" in astronomical and cosmological observations, where we deal usually with gravitation and electromagnetism one finds again the Lorentz manifold of standard GR.
This latter point of view is nicely summarized in
P. Ramond, Field Theory: A Modern Primer,
Addison-Wesley, Redwood City, Calif., 2 edn. (1989).
