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

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.