I thought the dialogue indicated that all flights were grounded due to the migration of the space creatures? My guess is that shuttles, both Starfleet and otherwise, are registered, but the loophole was that the Phoenix was seen more as an amusement ride for short bursts in near-space? Just guessing.
Well, shuttles are capable of getting you to other star systems (VOY's Type-8 and 9 shuttles could hit Warp 4) and have phasers, so there's potentially a lot more trouble if someone steals one. The Phoenix replica is limited to Warp 1 so whilst it's useful for pootling around the Sol system it's not going interstellar with the limited fuel and supplies onboard, even if someone did do what Mariner et al. did here.