Interestingly, I've been told there are indications that in the 24th century Starfleet keeps a Constitution-class starship "USS Republic" in service as a training vessel; something about the Connies makes them great teaching tools for cadets, maybe?
The "indications" come from the episode "Valiant", where even the ignorant civilian Jake Sisko has heard of the famous training ship Republic
. No class identity is given for that ship, other than her being at least fifty years old. Might be Starfleet is now in the habit of building dedicated training ships that all are named Republic
- indeed, the one from TOS "Court Martial" may have been a purpose-built training ship already, for all we know, carrying an already venerable name. Or then Starfleet for some other reason has chosen to employ a generic starship named Republic
as a training ship again in the 24th century.
Assuming, of course, that the TOS ship either was a training ship by design or served in a training role. There's no direct evidence of either: any regular frontline starship might have filled the dramatic role of the ship where Ensign Kirk serves "several years" after graduation. We only indirectly know that Kirk should still have been associated with the Academy at Ensign rank (supposedly as an instructor), thus the Republic
should have been somehow associated as well. But not necessarily as a training vessel.
I guess the only pressing reason to think that the Republic
of the 24th century is not
the one from Kirk's past is that Picard insists the Constitution
class only exists as a single hull in a museum as of the 2360s. But he may well be speaking of the unrefitted version, and the Republic
might have received a refit.
Or then the Republic
from Kirk's past never had anything even remotely to do with the Constitution
class. After all, we have no good reason to think she did.