Using terriers to dig out the fox after it's gone to ground is definitely a UK or British tradition and one that did not translate to the US.
If one of our hunts culminated in running the fox to ground, the huntsman dismounted while blowing "Gone to ground", praised and petted the hounds and fed them cookies, and then remounted and we went off in search of another fox.
The local hunt *might* kill up to 4 foxes a year, going out 60 - 70 times in a season. In all my years of hunting, I only ever saw/participated in one kill. Sadly that was an accident - the hounds weren't even running. A hound investigated under a coop while walking to a new covert and came out with the fox in his mouth.
The secret wish of most of the field and staff was that any fox that gave good sport would live happily and well and give us another day's good sport.