The Greeks were not adverse to studying topics outside of the classic axioms, for example neusis, conic sections, or Archimedes work on quadrature (which presaged calculus):

https://en.wikipedia.org/wiki/Neusis_construction

https://en.wikipedia.org/wiki/Conic_section

https://en.wikipedia.org/wiki/Quadrature_(mathematics)

https://en.wikipedia.org/wiki/Quadrature_of_the_Parabola

They just preferred the simpler axioms on grounds of aesthetic parsimony.

As far as I know, the ancient Greeks never thought to fold the paper. It has, however, been studied since the 1980's by modern mathematicians:

https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms

It can be used to trisecting an angle, an impossible construction with straightedge and compass:

https://www.youtube.com/watch?v=SL2lYcggGpc&t=185s

It's more powerful than compass and straight-edge constructions, but not by much. It essentially gives you cube roots in addition to square roots. You still need a completely different point of view to make the quantum leap the the real numbers, calculus, and limits:

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

https://en.wikipedia.org/wiki/Dedekind_cut

So ultimately I don't know if it would have changed the course of history that much.

> Folds are powerful. One can trisect or n-sect any angle for finite n.

Does that mean folding allows you to construct (without trial-and-error) an accurate heptagon, even though you can't with a straight-edge and compass?

Intuitively, that seems wrong, I would expect many of the same limitations to apply.

Akira Yoshizawa actually used origami in a factory setting to communicate geometric and engineering concepts.