There are a whole raft of nuclear rocket designs that do exhaust some of their fuel (to greater or lesser degree). In some cases it's a deliberate part of the design, in others not so much. NERVA for example suffered from fuel channel erosion which exhausted some fission products. They had made a lot of progress in solving this issue before the program was cancelled, but not entirely. Gas core fission, is much more efficient (and comes in a couple of versions), but we'd have a long way to go to ensure complete containment of fuel. As for neutrons, you need to keep them in a fission reactor to keep it going (otherwise your chain reaction fissiles out).
These designs also make use of the propellant as the coolant (so you don't require massive radiators for heat rejection). LH2 is preferred (not water) as if gives you the best exhaust velocity (it also isn't great at absorbing neutrons).
Most fusion rocket designs I've seen directly exhaust burned fuel as it's the most efficient way to transfer heat to the propellant and it's all light elements that you want anyways. Heating propellant with neutrons directly is horribly inefficient