I think you're right, and integer-factorization is often used in these examples as a process that is hard to do but easy to verify. There are plenty of other processes that could be substituted in, e.g. reversing SHA256 hashes, that would likely be even less tractable to the target audience.

However, if P = NP, there is no process that works here - there's nothing that is hard to do but easy to demonstrate, and therefore no zero knowledge proofs exist.

Actually, that's not true either. It requires the definition that all polynomial-time algorithms run quickly and all superpolynomial ones run slowly. This is not an accurate definition for all practical problem sizes and this is where the analogies all break down. Polynomial vs nonpolynomial is more interesting to complexity theorists than "how many years would this actually take with a fast computer".