Where do we learn that the E-E would hold flagship status?
As for Fleet Captains, we don't know much about this rank, or even whether it really exists. For all we know, its one and only canonical mention, in "The Menagerie pt I", merely establishes Pike as holding the exact same rank as Kirk, that is, (Star)fleet Captain (as opposed to generic captain such as Harry Mudd, or captain of a lesser rank such as Commander Ramart of the Antares
or Lieutenant Commander Dax of the Defiant
). Certainly no "fleet command" roles have ever been associated with this rank - its only known holder was famed for teaching cadets, nothing else.