While the historical postulate approach is entirely compatible with the modern symmetry approach, issues such as the one of parity symmetry you bring up, make it clear that they are not entirely equivalent in their precision.
If you postulate the physical laws are the same in all inertial frames and the speed of light is the same in all inertial frames, to be precise this begs the question of what the precise definition of an inertial frame is. This is one of those concepts we intuitively understand, but which would be very difficult to make precise. Starting from these principles, and mainly drawing from experience from electrodynamics, it would be very tempting to claim special relativity predicts time reversal invariance and parity symmetry.
Then when parity violation is later observed, how would one adjust their understanding to this? Abandon SR, or just absorb it into the definition of inertial frame, thus preserving SR? Again, to avoid these issues, if using the historical postulates approach an inertial frame would need to be defined precisely. This would likely involve a procedural definition, thus depend on the physics, and so one would have to be very careful not to reason circularly. And parts of it would feel very artificial, like demanding an inertial coordinate system was right-handed instead of left-handed.
Postulating a symmetry is much more precise. Does Poincare symmetry include parity transformations? No. Therefore SR doesn't predict one way or the other regarding parity symmetry. Simple, done.
Continuing along the idea of trying to absorb broken symmetries into the definition of an inertial frame, what if one found there was a preferred direction, but Lorentz boosts perpendicular to this axis was still a symmetry? Sure I guess if part of physics involved this "conserved vector", then we could still define an infinite set of frames which are "inertial" frames, move relative to each other, and physics looked the same. In the parity case, it would be, do you want to demand inertial frames using a right-handed coordinate system? Here it would be, do you want to demand that space looks the same in all directions? As you can see, answering such questions in trying to precisely define an inertial frame, is really just inching our way towards the symmetry case.
So while it would be straining credulity: it depends on how you define an inertial frame, whether we could obtain full rotational symmetry of physics from the postulates. Did you include rotational symmetry in the definition or not? And in that sense, it would not even be a derivation, it just has to be taken a priori.