The (non-generalized) Pythagorean theorem is part of Euclidean geometry, so non-Euclidean geometry is irrelevant to this discusion.
> Euclid's third postulate - "a circle can be drawn with any center and radius - doesn't define how to do it.
You do it using an axiomatic compass, a device that copies length in a circular pattern but does not measure it. Lengths are measured using constructable line segments.
Are you implying that nearly all the hundreds of proofs of Pythagorean theorem, which do not use modern rigorous definitions, are not valid proofs?
> Conversely, if you take the "conventional" definition, than the Pythagorean theorem falls out almost immediately.
So? The Pythagorean theorem is very easy to prove. There are hundreds of proofs created by amateurs. That doesn't make them "not proofs" simply because other proofs exist.
Strictly speaking, the postulates say nothing about compasses, or even straihhtedges/constructions. Also introducing lengths similarly, involves introducing number which is not a "pure" geometry concept. The third postulate just says that a "circle" exists defined by a point and a radii (which also, not a "pure" geometry construct since it involves a metric - i.e. number.
I would say yes, alot of the fundamental proofs while not striclty "incorrect" or false, are rather informal and contain some hidden axioms/circularities.
Tarski put geometry on a more secure footing using first-order logic.
Similar to how Calculus wasn't on a solid logical foundation until Riemann.