...And the only time any of the Enterprises
is considered a flagship, it is in one of these alternate senses.
The E-D is "Flagship of the Federation" in a couple of episodes and the movie ST:Generations. That's obviously not a military concept at all, because of the "Federation" rather than "Starfleet" bit there. Rather, the E-D apparently is the showpiece vessel of the Federation, being sent to impress the natives in various diplomatic contexts and whenever a bully needs bullying back; we see this happen often in TNG.
In contrast, Kirk's Enterprises
never were called the flagship of anything. Neither were the E-B, the E-C or the E-E, or Archer's pre-Federation starship Enterprise
. These vessels might have flown the flag of some admiral or another on occasion, or been used as kingpin vessels in starship formations, or paraded around in the "Flagship of the UFP" sense, and indeed many of them were, explicitly and on screen. However, dialogue never associated them with the flagship concept.