Certainly the flagship of that task force or cruiser group or whatever you prefer. But I don't see how that extends to a flagship for the whole of Starfleet. Shirley that would involve a higher-ranking officer than a commodore, and the best evidence indicates that Starfleet's top officer is based in San Francisco, not aboard a starship somewhere.
Indeed, I don't think the idea of a "flagship for the whole of Starfleet" is even meaningful. A flagship is the command ship for a jointly operating group of ships, and most Starfleet vessels operate alone. Too many fans -- and unfortunately too many Trek writers -- confuse the vernacular usage of "flagship" to mean the most prestigious or important member of a group (like the way Voyager
was UPN's flagship program) with its formal naval usages.
And don't call me Surely.