I never saw a leghold trap in my life. I have seen and worked with a lot foothold traps though. Modern footholds have offset jaws, padded jaws, swivels and shock absorbing springs all designed to keep caught animals from hurting themselves. I caught a lot of yotes and foxes in #3 Victor padded footolds when I was doing my research on coyotes and not one of them had any foot damage or problems running away if they were released.
Conibears BTW don't "instantly" kill many animals. If the animal is lucky it is knocked out or has its skull fractured. Usually it is choked or suffocated by the pressure of the bars not allowing it's chest to expand. This takes several minutes or longer.
The fastest killer I've seen of foxes and yotes has been a properly set snare (never used a power snare so I can't comment on them). We'd take about twelve feet of barbed wire and use it to anchor the snare to a tree and leave the twelve feet coiled on a peg on the tree with the snare set over the run. The yote would hit the snare, panic and go on a full out run -- for twelve feet. Most animals were found dead with absolutely no sign of struggle.