It's possible that the Enterprise and her sisters haven't been launched yet (I don't care what an encylcopedia says).
Me neither. But Star Trek says NCC-1701 is already flying around, and has been doing so for a couple of years already. It's 2256, and Pike's Big Adventure took place thirteen years before early TOS or what works out to 2266 at the very latest.
Also the constitution-class was more about deep space exploration
...And about guarding the long-silent Romulan Neutral Zone. What's Pike's excuse for not guarding the long-silent Klingon Neutral Zone? Is he racist?
so it'll be far away from the federation and even at max warp it'll take a while for it to get back.
But since Kirk did RNZ sentry, despite his deep space stints, it follows that
somebody probably is at the RNZ when Kirk is away, and that a perfect mirror in which this somebody flies a
Constitution is perfectly possible given the precedent.
Same goes for the Sovereign-class. By the time of the Dominion war there wouldn't be that many of them and most of them would either be protecting Federation member worlds or out in unexplored space.
Picard never took his
Sovereign to unexplored space. Would others? At the very least, since Picard didn't, his ship should have been available for the "protecting the UFP" bit. Many planets to protect out there - but apparently fleets of starships aren't involved in that stuff much, as the fleet near Betazed was in training maneuvers rather than minding the shop when the Dominion came.
The fleet protecting Earth refused to join the effort to liberate DS9, indicating Sisko believed it would be able to join in the time allocated. For most of the operations we see in DS9, the schedule wouldn't have been quite that tight. If taking Chin'toka really was important, why wouldn't the E-E be sent there? If taking Cardassia itself was, again why would the E-E mind a side theater (especially as the enemy had just volunteered to vacate the side theaters)?
I don't remember many/any Akira's either.
All the fleets had those, because the model was so pretty and detailed. All also had
Mirandas for the same reason, and then filler ships. The only ships that didn't make regular appearances were
Nebulas, again because the model at that point wasn't particularly good, but there were a couple of appearances nevertheless.
The total absence of the
Norway was due to the model being lost. The total absence of the
Constitution was due to the model not being easy to film, and to dramatic decisions about not showing "Kirk's ship". The total absence of the
Sovereign would have been purely due to dramatic decisions, though. (The total absence of the
Oberth was just common sense.

)
Timo Saloniemi