Poincaré was a genius but math has moved forward since the 19th century.
Here's what I'd argue is the most important take away from set theory: for a given framework of mathematics, there are conjectures one can make that can be neither proved nor disproved. Perhaps the most high-profile of them is the continuum hypothesis.
Maybe P!=NP is such a conjecture. Maybe the existence of a solution to the Navier-Stokes equation is such a problem. For Poincaré, the very idea that there is math that can be neither proved nor disproved was not in his mental vocabulary.
On another note, Weil's celebrated proof of Fermat's last theorem relies on the existence of inaccessible cardinals[1], although I've heard it conjectured that they are not necessary.