Folds are powerful. One can trisect or n-sect any angle for finite n. One still needs the compass though for circle.
Straight edge
Compass
Nuesis
Paper folding
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.
Origami folding is more powerful than the closure of rationale by square and cube roots.
They were extended to the quintic roots by Robert Lang using a type of folding called multifold. Now it's known that with multifolds all of the algebraic numbers can be constructed with origami
https://arxiv.org/abs/0808.1517
Yes one would not reach the reals (that's not the ultimate goal) but the geometry would certainly would have been richer.
By no means is the area of folding a mathematical dead end as new theorems still get discovered.
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.
https://origamiusa.org/thefold/article/diagrams-one-cut-hept...
The one cut is to remove the perimeter of the square that lies outside the heptagon. Without the cut, you could make a crease, and fold the excess behind the heptagon.
Certainly good enough for practical handheld construction purposes, but not geometric-proof-y stuff.
Scimemi, Draw of a regular
heptagon by folding.
Proceedings of the 1st
International Meeting of
Origami Science and
Technology. 1989
Are you familiar with Lill's method of finding real roots of polynomials of any degree ? Simultaneous folds are a realization of the same idea
https://en.m.wikipedia.org/wiki/Lill%27s_method#Finding_root...
http://origametry.net/papers/heptagon.pdf
It shows both a single sheet and a modular version.
But remember one is dealing with idealized / axiomatized folding. The situation is similar with compass and straight edge geometry -- those physical lines and circles marked on paper are approximate but mathematically, in the world of axioms we assume the tools are capable of perfect constructions.