You can invent and pick axioms in many ways that (probably) won't lead to inconsistencies. But they won't all be powerful enough or relevant in the real world.

Well, then it sounds like your reasoning gets it backwards: the axioms that produce systems without significant consequences or connections outside of their own abstract realm end up being ignored.

Or in other words: the constraints on maths are imposed from outside of maths.

Doesn't this imply that, while you can invent all the axioms you like, you must discover which ones are consistent with each other and with experimental results.