This just steps one meta level higher. Yes, you can make your object of analysis the axioms and what they lead to and proof theory etc. But now you've just stepped back one level. What are the axioms that allow you to derive that "ZFC leads to Hahn–Banach paradoxes"? Is this claim True and discovered or is it in itself also simply dependent on some axioms and assumptions?
This is part of a broader meta-ization of culture. Philosophers are also much more reluctant to make truth claims in the last century compared to centuries ago. Everything they say is just "To a Hegelian, it is {such and such}. For Descartes, {x, y, z}." If you study theology, they don't teach with conviction that "Statement A". They will teach that Presbyterians believe X while the Anglicans think Y, and the Catholics think it's an irrelevant distinction. Of course when push comes to shove, you do realize that they do have truth claims, and moral claims that are non-negotiable but are shy to come forward with them and explicitly only talk in this "conditional" "if-then" way.
In fact many would argue that math is not too far from theology. People who were obsessed with math limits, like Gödel, were also highly interested in theology.
I guess physics is the closest to still making actual truth claims about reality, though it's also retreating to "we're just making useful mathematical models, we aren't saying that reality is this way or that way".