I doubt starfleet would relegate the Enterprise from a front line exploration ship to a training vessel without good reason. Why waist a Constitution when they could have a survey ship instead?
I got the impression that rather than the Enterprise being a "general purpose" training vessel, the purpose of the training mission specifically to train a new crew for the Enterprise. This is based on McCoy's comment to Kirk after the Kobayashi Maru simulation, "Admiral, wouldn't it just be easier to put an experienced crew back on the Enterprise" (or something to that effect).
They weren't training cadets to serve on other ships, they were training them to replace a previous crew. There is no need to assume that the Enterprise had been downgraded or taken off exploration assignments for good. She was still ship of the line, they just in the process of familiarizing her new crew with the ship and systems. Also, Kirk indicated he wanted to return to the Genesis Planet at the end of TWOK. Not a likely expectation if the Enterprise would only be doing training missions for the foreseeable future.
What is uncertain is what would have happened to the command staff once the crew was prepared for actual duty. Kirk had taken himself out of the equation. Spock didn't seem to want command outside of a teaching environment (of course he might have just been trying to encourage his friend assume command). Sulu seemed to have another assignment. It's entirely likely that a new commander would have been brought in.