No, out of universe TMP and TNG cannot be held responsible for a TV show that came out decades after the fact. On the other hand, naming the NX-01 something other than Enterprise solves the continuity error. It's not hard. In fact, naming it Valiant might have actually been a nice continuity tie-in.
We're Star Trek fans, though. We don't care about real-world practicality. In universe the NX-01 should have been included for it's historical importance. The Enterprises that were depicted in TMP were significant historic ships named Enterprise. As depicted in ENT, the NX-01 was just as historically significant as these other ships named Enterprise.
This highlights one of the real problems (I have) with retcons and prequels. They get painted into corners and are either "stifled by the creative restraints" or they ignore continuity.
From the Cage I got the impression that Number One was both First Officer and Helm Control.
From Where No Man I got the impression (and I think I was wrong) that Gary Mitchell was First Officer and Navigator.
It was clear from TOS that Spock was First Officer and Science Officer
My conclusion was that First Officer was not a distinct and separate position from any other bridge position. The First Officer would be one other position as well.
It wasn't until TMP that the First Officer became distinct and separate. With the death of Xon in the transporter accident, Decker had to double as First Officer and Science Officer.
Riker in TNG was the first character to exclusively be First Officer.