The TNG writers did take care to describe the E-D as the Federation flagship, that is, the flagship of the civilian political entity. What role the E-D played within Starfleet was left unspecified...
The ship was described as an "explorer", which may have been a generic expression or then a futuristic designation equally specific to today's "destroyer" or "corvette". She was also shown serving as the kingpin of a fleet or other formation on occasion. But did any flag officer ever unfurl his or her flag aboard? Nechayev just visited the ship, and this doesn't make the E-D a flagship in the formal sense yet. Pressman tried to take direct command of her but failed, yet he was in command of a mission and potentially "broke his flag" for the purpose, assuming Starfleet speaks in such terms and follows such practices.
We also have to consider that future flag officers might make specific ships their "virtual" flagships, what with physical presence being more difficult and virtual being easier than today... Perhaps the E-D was formally the flagship of Admiral Haden in all those Romulan episodes? (Well, probably the worst possible examples of such, considering the comm lag there.)
...Whoops, this wasn't the E-D -specific thread ongoing in that other sub-forum.
Regarding Kirk's ship, yeah, I'm sure Starfleet would take great care not to claim that she was their finest flagship and the one every propaganda piece and press release should be about!
Timo Saloniemi
The ship was described as an "explorer", which may have been a generic expression or then a futuristic designation equally specific to today's "destroyer" or "corvette". She was also shown serving as the kingpin of a fleet or other formation on occasion. But did any flag officer ever unfurl his or her flag aboard? Nechayev just visited the ship, and this doesn't make the E-D a flagship in the formal sense yet. Pressman tried to take direct command of her but failed, yet he was in command of a mission and potentially "broke his flag" for the purpose, assuming Starfleet speaks in such terms and follows such practices.
We also have to consider that future flag officers might make specific ships their "virtual" flagships, what with physical presence being more difficult and virtual being easier than today... Perhaps the E-D was formally the flagship of Admiral Haden in all those Romulan episodes? (Well, probably the worst possible examples of such, considering the comm lag there.)
...Whoops, this wasn't the E-D -specific thread ongoing in that other sub-forum.

Timo Saloniemi