Mouse/rat traps are meant to kill , and usually do effectively
The leghold traps will not break the target animals leg or kill, The early versions pre 1960's were designed a little differently and could cause the animals legs skin to break but not always. Even when this happened the animal surely was um-comfertable but not realy in pain, Most of the time when the traps are checked the animal will even be asleep (even with the early ones). The most modern versions of the legholds have padded jaws that are very humane. Their are even videos from makers of some of these modern traps showing a man step into the largest stock traps they make
The worst think that can happen to animals in the traps is much smaller animals geting caught in them, Much more likely is the animal doing damage to their own leg trying to escape, Or another animal taking advantage of a trapped animal.
One more thing that is worth a mention is predator urine/scat . My neighbor has a animal that fits the description and I have not seen a fox since they moved in. Before they moved in I caught a unbelievable amount of them, then nothing for about 3 years now