SuZir, the type of hunting you describe: using "earth dogs" to dig the fox out of its hide to be shot, or trapping it an shooting it, are standard methods of control in many areas, though not the most humane.
They are also accompanied by tradition.
"Earth dog" competitions were popular in the area of Germany in which I lived, with terriers and "teckels" (Dachshunds) competing.
Tunnels were dug maze-fashion, and a rat in a cage was placed in a chamber at one end of the maze.
The dogs were placed at various entrances to to the "maze". The dog which reached the rat first won the competition.
Neither rat nor dog was harmed during these competitions, though I imagine it had to be terrifying for the poor rat.
I had a longhaired teckel for a while in the US, and what he did to my back yard when we acquired moles was truly terrifying. I have NEVER seen a dog dig like that!
The problem with the British views on foxhunting go far beyond the cruelty to the prey, however. It is sport associated with the elite; the "landed gentry", and it hearkens back to a time when the poor weren't allowed to hunt in the great forests that covered much of Britain on pain of death.
A British writer or playwright, I cannot remember who at the moment, once characterized foxhunting as, "the unspeakable in pursuit of the inedible"., which basically sums up the bulk of public opinion in the UK.