I agree that they could have existed alongside with the ship classes of the NX-Project.
Absolutely. There were only, what, six ships in the entire NX class (if you take literature into consideration)? That's hardly enough for an entire fleet. And since the maximum speed of the other ships - the Delta and Intrepid types - was probably very low, like maybe Warp 2, I don't see how they could have comprised the whole fleet either. I don't see any discontinuity whatsoever in the Museum ships existing alongside the ENT ones.
That being said, I think the Museum ships look really cool (AFAIK, the reason they look the way that they do is that their decks are all facing forward - meaning, 'top' on these ships is actually pointing forward) and I'd love to see what they'd look like onscreen.