[Q-Q]

I find it not believable that

- You derived the theory correctly (end result should match textbook result) and thought what you were doing were wrong.

- No physicist you asked could explain where you were wrong

- No physicist you encountered could see the obvious connection of this and the usual approach

It's not really a mysterious or novel fact that you can have these shift of perspectives (at least since Kant who heavily influenced Einstein) e.g. Newton took his three laws as the base, derived relativity as a theorem. Einstein took (modified version) relativity (together with the invariant speed assumption which comes from electromagnetism) as the base and rebuilt the rest of theory of motion that reduces to Newton's theory in the limit. Nothing is amiss here.