Actually, here in the United States, with the notable exception of the Eastern gentrified horse country people, most of us are much too lazy to attempt running down a fox, deer, or hare on foot, on horseback, or by vehicle. However, we are permitted (depending on locale and applicable laws) to stalk and kill them suddenly and instantly with bow and arrow, shotgun, rifle or handgun (and by bear spear, if you are Ted Nugent).
Probably it is the difference between wanting to preserve the animal's utility (as pelt or meat) as opposed to running them until they are exhausted and then letting them be ripped to pieces by the dogs for "sport." (Although I have to acknowledge that dogs are used in some parts of the country to run down and tree certain types of game.)
What I was trying to ascertain is if the law still permitted the taking of these animals (presumably by firearm) if dogs were not used in the process of stalking and killing them. Or has that type of hunting pretty much been banned in the UK as well?
. . .running them until they are exhausted and then letting them be ripped to pieces by the dogs for "sport."
In the United States, foxes are not run until they are exhausted and then ripped to pieces by the hounds. They are run for a few minutes until they decide they're bored and choose to go to ground in a hole or, in the case of gray foxes, climb a tree. It is rare indeed that they are killed. The only ones who die are extremely sick, usually dying of mange. And when they are killed, they're killed by a quick snap of the neck.
In the UK foxes have no predators or illnesses like rabies to reduce their numbers, so they have become a very serious agricultural pest, a predator that decimates poultry and devours lambs-aborning in the fields. Farmers in the UK welcome foxhunts because they can't deal with the foxes on their own. British agriculture is already struggling after their recent bouts of hoof-and-mouth disease, which represented an economic disaster, and the last thing they need is more predators eating their stock.