Seems that some folk haven't quite understood how British wildlife is grouped as such. The easiest way I find rather than to go trawling through gov websites for laws and arguing over them is simply read the hunting act. It clearly explains how the majority of all wildlife in this country is broken down to three groups.
Protected species: These are fully off limits to anyone and severe penalties await those who ignore such.
Non endangered wildlife: Here lies most of the wild animals in this country, they are not in danger and also not at full pest levels as such they are wild animals and to be left alone. In certain situations localised populations can become a pest problem and approval for control methods can be sought for that local group only. This group also tends to include in season species such as ducks etc.
Pest species: Here the population numbers have become a problem to the extent that any legal humane means may be used to control numbers at anytime,this group includes your foxes, pigeons etc.
I'm willing to stick my head on the block here and venture that 99% of all folk who complain about country folk controlling species on their land are people who live in towns & cities. They also probably practice pest control in their own homes & gardens upon bugs/slugs etc with ruthless determination, finally they probably happily go to the local supermarket each week to buy an animal part for a sunday roast without any major thought for how it came to be there.
Compare that to the true country folk who watch the changes on the land through the years and you get a real educated view on population numbers that a lobby run survey can never reveal. If farmer Giles tells me the foxes are out of control I'll believe him before any report commisioned from from any animal welfare organisation who sends one city dweller out to the fields for a weekend with a ticker.
Folk happily complain from comfy armchairs about how terrible it is that a bird was shot or a fox trapped etc, but then want more and more food for an expanding human population that physically forces farmers to sterilize the fields, if they do not do this the crops will not provide enough output. Yet take a stroll in any managed heath or woodland for country sports and it's heaving with wildlife.
To argue that country sports do nothing for wildlife protection is ludicrous as many species exist almost nowhere else but on sport type land. If the sports were ended that land would go one of three ways. Sterile farm fields for crops, Government body managed forests used for pretty much nothing due to draconian laws about them (can count the species in these on one hand), or simply paved over for development.
Someone pointed out we have few true apex predators like golden eagles etc in the wild to compete for food, this is a harsh fact that is unlikely to change with human numbers and land needs. As such we need to take that role and keep a balanced number of other species, simple fact is that requires killing, culling, controlling or whatever name you wish to give it.