I always found it puzzling they only ever referred to one flagship in TNG.
But they didn't.
The Federation might have had only one flagship. Starfleet had several, though:
Adm. Nechayev: "The Gorkon will be my flagship."
In time-honored tradition, whenever an Admiral took command of a starship formation or an operation, she or he adopted a specific starship as her or his flagship. Like said above, Admirals Hanson and Hayes had specific ships they used for commanding their fights against the Borg. Those would only be flagships as long as the admirals utilized them as such, of course: any random ship could be put to flagship use at any moment, although some ships would no doubt be more suited for it than others.
I'd disagree on the original
Defiant ever having been a flagship of any sort, though. Admiral Ross never set foot on her AFAWK, and she was never referred to as anybody's flagship. Captain Sisko sometimes told other starships what to do while sitting on the command chair of the
Defiant, but there's little reason to think other skippers didn't do likewise, without their ships becoming any more "flagshippy" than the
Defiant was. Sisko may have planned a few big fights, but their actual theater command may have been handled by proper flagships with trained flag officers aboard them.
The second
Defiant never had Admirals aboard during missions, either, nor did she participate in missions where Sisko would have commanded starship formations or even been credited with planning the attacks.
Timo Saloniemi