So you say fox hunting is necessary to maintain a stable population. But what evidence is there to show that the population would become unstable if foxes weren't hunted? I am not aware of any, and it isn't what one would expect given what we know about fox ecology. Rather, all hunting does is reduce the overall population from a stable level to a slightly lower stable level. Isn't that true?
Ok, a question asked out of ignorance: If a fox has been taking lambs or has broken in and killed hens, isn't it possible to wait for it to return, lamp and kill it? My guess is that you'd have to be very patient - perhaps too patient for it to be an economically viable solution. How about going out lamping in the area where the problem occurred? If you shot a fox, you'd couldn't be sure it was the 'rogue', but couldn't you carry on doing this until no more foxes turned up? Of course, in time another fox, or other foxes, would invade the vacated territory - but that would happen no matter what method of killing was employed, and the replacement fox may not be a 'rogue'.
As I said before, I am against the use of snares (and traps that injure). It sounds like you are too...?