I would say, there are known sub-exponential algorithms for factoring, while nothing better than (exponential) brute force is known for ECC. So, slightly better algorithms that erode RSA further are incremental improvements, while anything sub-exponential at all for ECC would be a breakthrough. Of course as someone once said, predictions are hard to make, especially about the future.

Four decades ago, the professor who taught me P vs NP absolutely stuffed his explanation of the polynomial refactoring example, drawing sum of products on the board as his example - bugged the crap out of me because I could visualize code for an algorithm to refactor sum of products relatively easily - certainly in polynomial time. I brought that to him after the next lecture and he clarified: product of sums is the hard problem, well, yeah, obviously when you look at that nightmare. I assume he had been teaching it for about a decade as well, he certainly had been in the department that long and longer.

You wrote "we know for certain do that it [factoring] is not in NP hard". That sentence is 1) borderline ungrammatical (we usually use NP-hard as an adjective, though it also denotes a set); and 2) in error. We suspect factoring is not NP-hard but we don't know for certain (consider what happens if P=NP). The error suggests confusion about what these terms mean. If you're really teaching this subject, can I ask what textbook you are using?

Factoring is in a category. Everything computable exists somewhere. We know factoring to be in NP.

What I was disagreeing with was the "at least" and "at most" characterization of NP completeness. It is a set, not a boundary. The actual diagram of the complexity zoo is much more complicated than concentric circles.

And for verifiability, I was not referring to it as an existential sort of thing. I was simply saying that I agreed with you in that particular facet, but it isn't sufficient to describe NP completeness. You also need NP-hardness.