Stopping and watching doesn’t make it a game. For the wild creature to stop and watch means they are making a decision whether the threat is oncoming and for them to continue running to save their life, or the threat has turned and they can relax and go on about their business. Young ones usually keep running because instinct tell thems to, and they have a lack of worldly knowledge to make an informed decision on whether they can safely stop… or not. Old ones, who have lived long enough to understand the cause-and-effect actions of other preditors, will pay attention to the sights and sounds of an event and determine when(if) the chase is moving away from them and (if) they are not in the line of fire. They will wait, watch, and listen. But this is NEVER a game.
The fox does NOT know that the hounds are after it unless the pursuit gets too close and too personal (the sound continues to advance and stay strong in the direction from which the fox has traveled). All wild animals will scatter at the sound of hounds coming on --rabbits, deer, racoons, fox, cats, birds, etc – they only know a preditor or pack of preditors is on the loose… and if the echos of sound are clear and growing, they know also they are standing in the path of approaching danger, and need to move. The sound alone is enough to warrent any creature into running, which is why hounds were bred to produce the long, melodious yodel – to keep the game running. Silent hounds are culled, or not bred back into the gene pool.
It is only when the threatening sound waves reduce or move away that a creature will slow or pause. Instinct will tell it at that point it is safe and can stop running. But if the sound returns (in strong or increasing volume) in the direction in which that creature is standing, the flight instinct kicks in again.
Where you’re wrong is the analogy to the gladiator fight (no escape for the glads out of the arena) and the bull fight (ditto: not only no escape for the bull out of the arena, but first it’s stabbed and wounded and weakened by mounted horsemen with those spears before the matador even enters). So these are terrible and totally inaccurate analogies to a free-ranging wild animal. Not even close.
I used these two merely to illustrate the “sport” of pitting of one creature against another for the amusement of spectators. I didn’t compare them to each other, nor did I desire to.
So the question is are the ones who don’t like it allowed to decide what the ones who do like it can do?
No, the question that was asked was: do the hounds kill the fox? The answer is: yes, if they can get it.
The rest of your post is a tangent based upon human moral issues – not part of this discussion. Nor will I go there. 
PS. I noticed it was also asked “Can somebody tell me that fox hunting in 2006 isn’t quite the same as fox hunting of old?” Answer: in America it is pretty much identical in form to British hunts…but not function. It was never an American practice to “stop” the earths (plug up the known dens of foxes in the area that would be hunting the following day) as in Britian and Ireland. Also (nowadays) American hunts do not chase to kill. If a fox goes to ground the huntsman will “honor” it --respectfully salute-- and leave the creature and site in peace, taking the pack with them. The focus for American hunts (today, although not up to about 50 years ago) is on the find and the chase, not a kill. Other than that, the general acceptance of dress, social protocol, structure, breeding of hounds, people hunting, horses used, etc all remains pretty much intact from the prior century. There are quite a few pretty hilarious stories one can read on hunting of old – shows that very little has changed for some people even today!
