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):
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:
Sure, it makes sense to isolate the minimal sets of primitives needed for an operation. Greeks experimented quite a bit with nuesis before focusing on straight edge and compass. Folding, as you noted, was not part of their mix. BTW nuesis can also trisect angles, so they could do it without origami.
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
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.
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.
Folds are powerful. One can trisect or n-sect any angle for finite n. One still needs the compass though for circle.
Makes for a very powerful tool set.