The USS Minotaur would be between 100 to 75 years old but her class is still a reliable workhorse

We can always say she has just undergone a total refit after being heavily damaged in some battle with a minor race's fleet; It took a fleet to damage her 6-12 ships at once which means she as a ship has her own glory and perhaps even an Admiral as her patron who served aboard her or commanded her during the Borg War or even the Dominion war

Because she was in the Refit Yards for say 2 years her crew was needed to fill places in the Fleet meaning she'd need a new crew from the ground up

I actually agree with the Department head part for Bridge crew but feel it necessary to say that on a ship with 700-1000 crew it is common that the first officer is a Lt. Commander or perhaps a Full Commander but our ship isn't exactly a front line ship at her age. Because of that I can see her having Lieutenant JG as a department head placed there to season them for promotion; Geordi was appointed CE he was raised to the Rank of Lieutenant Commander but the Enterprise D is a newer ship and much larger than an Excelsior class.

I hope I don't come across as a pain but I am a hardcore Trek Fan and have been running a LUG campaign for 8 years now and love to read the novels so I've picked up a lot of details; even now I am reading the Typhoon Pact books and while not canon it is a good read