These discussions are only going to go one way, because the people who understand the issues are the ones who live with these species on a daily basis. Farmers hate foxes and grey squirrels for good reason, and whether we like it or not they'll give them two barrels on sight.
It's all too easy to take the 'Don't shoot that furry critter' stance when you don't have to live with the problem. What doesn't help matters is that the closest people come to the killing of the food they eat these days is that frozen shrink wrapped package they get from Morrison's on Saturday. We now have several generations of people who have never, and will never have to, kill and draw a chicken. We're all so bloody civilised, but the realism is that someone else is doing your killing for you by proxy.