On elementary proofs of Fermat's Last Theorem

I think the claim of that thread is blatantly overstated. For one thing, there are lots of properties that $\mathbb{Z}$ has, but $Z_p$ does not have. First and foremost, well ordering of the positive elements, which is heavily used as a key to solving many diophantine equations.

Now consider Fermat's elementary proof that $$x^4 + y^4 = z^4$$ has no solutions for $(x,y,z) \in \mathbb{Z}$ with $xyz \ne 0$.

I'm not sure whether or not there is a solution in $p$-adic integers, for some prime $p$, but if such solutions exist, it's an example showing that the existence of qualifying $p$-adic solutions doesn't imply the existence of qualifying integer solutions.

As I indicated in the comments, I suspect that most flawed attempts, at least the ones where the solver knows some Number Theory and is not obviously crazy, would include steps for which there is no analogue in $Z_p$, so the $Z_p$ criterion would be useless for invalidating those attempts.

For a given proposed proof, a more common way of quickly demonstrating that there must be a mistake$\,-\,$without actually pinpointing the error, is to observe that the argument would still work for the equation $x^2 + y^2 = z^2$, or, alternatively, to apply the argument line by line for the equation $x^3+y^3=z^3$, and see if the proof at least works for that case.