Then again: JJ. Abrams original plan was to blow up the original Enterprise at the beginning of Star Trek (2009), and not the "USS Kelvin", and he was straight up tolled "No" from Star Trek's owners, that he was not allowed to do that.
Also, he allegedly wanted all prime universe merchandise to stop being produced (Shatner- and Nimoy- alike figurines and stuff) to avoid "brand confusion" and was - again - stopped from it.
Also, the original constitution class model that appears in one of the deleted scenes of "Into Darkness" is (IMO!) also at least 25% different from the very original Enterprise - so they might not be the ones owning the original Enterprise either.