Every Enterprise in Star Trek has made for an exceptionally poor Flagship for the following reasons:
1) No flag officer was ever intended to command the Enterprise. (RAdm. Kirk stole command from Capt. Decker, remember.)
2) Enterprise has never had any involvement in the command and control functions of Starfleet. By definition, a flagship is meant to coordinate a fleet, not wander the cosmos LY from the nearest escorts.
3) A flagship would, as I said before, be surrounded by escorts. Starfleet would be out of their damn minds to send their actual flagship on five-year missions of exploration.
4) Flagships are not always the fastest, newest, or most powerful ships in the fleet, whereas all Enterprises, except the Ent-A, have been named Enterprise because of their speed, novelty, or power.
That said, I would expect the true flagship of Starfleet to have a separate 'flag bridge', for use by the Admiralty and their staff, as well as a command bridge, and a few other modifications (faster warp drive, maybe).
My conclusion is that Enterprise is not Starfleet's actual 'flagship' (as in, a mobile command and control center), but simply the most well-known ship and Starfleet's icon of power.