Did you even read his post? what you are calling the "right" path is the "mathematician" path, assuming relativity + group structure.

What he complained about, the Physics textbook approach was the path taken by Einstein who started from the "invariant speed" assumption to derive relativity:

https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/...

>"introduce another postulate... namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body."

There's also nothing really "wrong" with Einstein's approach either, it's motivated by physical constraint from Maxwell's electromagnetism.

Mathematician's approach is motivated by imposing logical constraint of an abstract mathematical structure (group of transformations from one coordinate system to another)

The two turn out to be connected (physical constraint selects a math model from alternatives). Nothing really strange.