^ The term "flagship" has been incredibly misused after TOS. The Enterprise was only a flagship in TOS, TWOK and TSF -- that is, an admiral commanded it. No admiral? No flag.
Nope. She was "just another ship" in TOS, no more a flagship than any of the other 11 Constitution class starships. In TWOK she was a semi-retired training vessel - FAR from a flagship. They didn't start throwing around the flagship moniker until TNG, and even then it was used incorrectly, since a flag officer was not in command.