I get the impression that Starfleet used them more like galactic troubleshooters rather than have them on general duties.
My pet theory, based on what we see on screen, has always been that following the Earth probe situation in STIV, Kirk and crew aren't exactly on the 'active duty' roster. On both occasions that we see the 1701-A, the crew are initially seen/said to be seperated from each other, and it's only a unique crisis situation which sees them all being recruited for a mission. I hold that this is pretty much it: the Excelsior was still undergoing tests in STIV and STV, hence why Sulu is still hanging out waiting for her to become available and is able to come back to 1701-A for the Nimbus II mission, and 1701-B is obviously on the drawing board as well. 1701-A is not an 'active ship' and incidents like the crisis at Nimbus III or the Khitomer accords see Kirk and crew being wheeled out, almost ceremonially, because they've got the peculiar skills needed for that one mission, and then afterwards they probably all go back into semi-retirement. Scotty talks about such in STVI, McCoy has no idea where Sulu is (and the others all act like they haven't seen each other in years
), and Spock is away doing diplomatic stuff.
I understand the need for fandom to fill in the gaps, but I've always thought maybe the 1701-A Enterprise had at best only a small handful of "special assignments" during that period, but was otherwise a ship that just didn't get out all that much.