Starfleet Academy would want recruits to be familiar with a variety of (types of) ships/tech, regardless of whether they'll ever use one after they graduate. So maybe they get to use the inventory.
Why assume the museum is only for Starfleet? Presumably the vast majority of its visitors are civilians. Academy cadets would train using actual Academy resources.
What a museum means by the late 24th/early 25th century doesn't have to be a static collection of non-functioning exhibits and realistic/tactically useless simulators.
I don't care what century it is, a museum wouldn't have active weapons systems in its exhibits.
Besides, the 24th/25th century has holodecks which can create perfectly lifelike simulations of the experience of using the ships, which means they'd have far less reason, not more, to risk damage and degradation to historically important exhibits by letting the public take them out for a spin. The purpose of a museum is to preserve important artifacts, and that means hands bloody well off.