^ Bridge crew, then. And yes, we all know that ships' crews don't stay together like that in reality. But it if was all a PR stunt, why did they send them to deal with the hotspot on Nimbus III?
As for the ships not being replaced with new ships of the same name, well, even if you disregard Jonathan Archer's ship, Starfleet went on to have Enterprises B-E (and beyond, according to Enterprise's third season). And the Defiant was also replaced with a namesake, when destroyed in the Dominion War.