Godel numbers remain brilliant - too brilliant, which is why nobody is using them.
Godel's proof includes a particular number, g. This is a specific number, like pi or e, albeit finite. Except it isn't. Nobody has calculated g. You can't, because the number is self-referential and doesn't converge.
Likewise Turing's halting proof diverges. You can write the formula for its godel number, but it is not computable.
Basically this happens because they made grammatical errors. In reality, being provable from the axioms is what it means to be true. Godel appears to prove that true things aren't true, which means it must be appearance only, not substance.
Someone really should have noticed g diverges. But they also should be using godel numbers in real applications, so...
As always, counter confirmation bias by looking for counter-evidence rather than looking for confirming evidence.
Nobody can point to a single true mathematical statement that is unprovable. The denial evidence does not exist, as expected.