It is no more or less illegal to cap them with something capable of killing them humanely than it is a .22lr.
The situation dictates a lot.
In a case of exigent circumstances like the video, I'd be shooting with whatever pistol I was wearing - so almost certainly 9mm.
In a case where I was illegally killing them either to prevent an attack or in retribution for one in a suburban or otherwise non-permissive area, I'd probably use something quieter like a suppressed .22LR or an air gun.
If I were trying to kill coyotes at my current house I'd either be using my carry pistol (again, probably a 9mm) assuming an unexpected encounter, or a centerfire rifle assuming I was actively hunting them.
I would definitely feel bad if I sent a canid (or any critter for that matter) off to die slowly, but if acting in defense of my family and animals - or even my neighbor's herds - I'd be willing to risk that.