My point is that axioms "feeling obvious" is exactly a signal that they will be useful. The point of deductive reasoning based on axioms is that it is a shortcu

My point is that axioms "feeling obvious" is exactly a signal that they will be useful. The point of deductive reasoning based on axioms is that it is a shortcut to fill in problems of induction, which is what happens when we use pure empiricism. If you really want to go down the road of solipsism, read Karl Popper.