The ship doesn't seem to enjoy any sort of special status (at least not a positive one) in ST5:TFF. Her role as a diplomatic platform in ST6:TUC might be more an exception than a rule: the first time in decades when it would be to Starfleet's advantage to send Kirk and the Ship of Kirk to a mission of diplomatic delicateness.
Certainly any ideas on a "flagship" status for the E-A (or indeed for any Enterprise
other than the first TNG hero ship, and the alternate behemoth from the latest movie) are more speculative than canonical. No Admiral is known to have set foot on the E-A, either...