Well, for a start, we shouldn't have to manage the population. Fox hunting doesn't kill enough foxes to really count as pest control, infact I bet there are ten times more foxes killed by cars and disease than by hunting.
Managing, or, culling certain species of animal to control the population is a very slippery and controversial slope. For a start, it's an eco-system, which means every action has a reaction. For example, foxes numbers go down, what happens to the rabbits, or badgers, or other animals that are either competitors to or prey of the fox? You mentioned the Grey Squirrel hunters, employed because of mistakes made in the past when introducing new species to a system. I could also point you in the direction of the Chinese government that killed millions of sparrows which resulted in millions of humans dying of starvation. I'm not saying it's the same thing, make sure you realise that, I'm just trying to illustrate a point. Infact I'm sure you know this, which is why I'm baffled as to how someone can honestly support something so fecking twatish.