The primary failsafe mode for an NTR would be to insert control rods to stop fission. Without the hydrogen the core wouldn't be able to cool itself and would melt down. There are however NTR core designs with closed circuit cooling. The core would be kept at a low critical state (hot but not melting) and circulate a coolant through the core and into a generator and from there to radiator panels. When the NTR wasn't providing thrust it would provide electrical power. When thrust is needed the coolant loop would cut off and hydrogen would be pumped through the core. Provided no mechanical breakdown in the coolant/generator loop an NTR could provide power for years.
I guess one could still build single burn/single use reactors. The thing would be a bit lighter than one that can survive multiple burns & it should not pose a Hazzard as long as you plan the resulting orbit of the discarded reactor accordingly.
For a Hohmann transfer orbit you need at least two burns, the perigee burn to put you into the elliptical transfer orbit and the apogee burn to circularize that orbit at your destination. Even free return trajectories can require a secondary burn. So in many situations throwing your engines away is not a great idea.
An NTR can be designed such that the engine and spacecraft "chassis" are reusable over multiple missions. NASA has/has an NTR concept with such a reusable vehicle. The fuel tanks are disposable and slot into the central frame like AA batteries. The crew portion would be a TransHab-like habitation module with a docked crew capsule and Mars lander. Propellant tanks would be disposed of during the mission and the vehicle parked in Earth orbit between missions. For a new mission propellant tanks would be fitted along with a new crew and off it goes. It's an interesting design but a little passed the current bleeding edge of in-orbit construction.
