The way formalists (mainstream mathematical community) dealt with the foundational issues was to study them very closely and precisely so that they can ignore it as much as possible. The philosophical justification is that even though a statement P is undecidable, ultimately speaking, within the universe of mathematical truth, it's either true or false and nothing else, even though we may not be able to construct a proof of either.

Constructivists on the other hand took the opposite approach, they equated mathematical truth with provability, therefore undecidable statements P are such that they're neither true nor false, constructively. This means Aristotle's law of excluded middle (for any statement P, P or (not P)) no longer holds and therefore constructivists had to rebuild mathematics from a different logical basis.

The issue with Principia is it doesn't know how to deal with issues like this, so the way it lays out mathematics no longer makes total sense, and its goals (mathematical program) are found to be impossible.

Note: this is an extreme oversimplification. I recommend Stanford Encyclopedia of Philosophy for a more detailed overview. E.g. https://plato.stanford.edu/entries/hilbert-program/