For me this proof was always a cautionary tale about being careful about recursion when you use any language.

It's the same thing as with set theory. Once you let yourself talk about sets of sets without any restrictions other than the language itself puts on what you are talking about, you'll end up with sort of contradictory recursion that ends up in a paradox. The message is not "set theory is incomplete, or invalid" but rather "we were a bit too cavalier with words and not everything we can say about sets makes sense just because it's grammatically correct so we need to be more careful about defining what a set is and what it can and cannot contain".

For me "provability" is a direct equivalent of unrestricted sets of sets. If you don't restrict what provability means and how it can be used in context of the rest of math you end up with contradictory recursion that makes you believe there are true but unprovable things. It's easier to spot how bonkers it is on sets because there you end up with a set that is and isn't its own element at the same time (because sets are so generic that can produce paradoxical recursion on themselves without any other concept involved), but things being unprovable and true at the same time is the same category of absurdity.