For TNG, it was more they needed a ship for those encounters and the Excelsior and Klingon models were available for no real extra expense. The Stargazer was probably the first unneeded, but welcome form of new ship...they were going to just use a redress of the movie Enterprise-A for that ship.
Those are all behind-the-scenes concerns. If a similar scene occurred in the first season, with a crew transfer at a starbase, and an NX-class ship slowly orbiting the station, would it really elicit the same kind of groans as C3PO & R2D2 in Rogue One (the ultimate in fan wank)?
But let's also step back a second and realize Trek has made a business out of what we now call "fan wank." It was the first true "universe" of shows, and so little references, both visual and verbal, to connect various stories have been happening since at least the premiere of TNG.
The real question is, do you think they would do it, and do you think it's reasonably plausible for NX-class ships to still be in service?