Nope, it would just report to a different base and have to deal with a Federation that's less human centric. It's not like they'd suddenly wander away. If San Francisco were destroyed in an earthquake, the ship stationed there wouldn't just wander the pacific. They're relocate to San Diego or Hawaii.
That would make sense if we'd ever been shown Starfleet as having the same presence on other planets that they do on Earth. Vulcan is one of the Federation's founders, but Kirk & Co. still had to go all the way back to Earth to stand trial.
Doesn't them destroying Vulcan, killing Amanda, reconciling Spock & Sarek years earlier than the Prime Universe and making Spock and Uhura a couple prove that that's exactly what they're not doing?

