Included in that handwave about probability distributions was a caveat about not being handed pathological instances of problems. Encryption harnesses pathological instances and wouldn't be covered by such a handwave.

(Is there ever any other reason to generate multi-hundred digit numbers and try to factor them? Honest question, if anybody's got a fun answer, though I am asking about something actually useful that you know about, not something hypothetical.)

Factoring numbers is already suspected to be much easier than NP-complete.

That is because factoring prime numbers is in the intersection between NP and co-NP. Co-NP is the set of problems where a "NO" answer is easily checkable with a certificate. If NP ?= co-NP is as much a question as P ?= NP. (I.e. no known proof either way, but every expects them to be unequal.)

Public-key encryption schemes would simply need to avoid reasonable circumstances. I suspect "product of two large primes" falls outside the "reasonable circumstances" for factoring.