From experience I can tell you that I have yet to see a deer that can't out run dog.
A push with dogs generally lasts at the most 2 hours in the bush. Of that 2 hours, the "chase" period where the dogs are running the deer typically lasts 5-10 minutes. The dogs simply keep the deer moving and provide the hunter an opportunity to line up a successful shot.
Once the push is over, whether successful or not, the dogs are collected and the group leaves the area.
Again from experience I can tell you that I can not tell the difference in meat taste between a dog run deer or a bow hunted deer or a still hunted rifle deer. And yes I have done the comparisons. The myth that the adrenaline spoils the meat seems to ME to be just that, a myth.
Everyone has different hunting practices, still hunts, stalking, baiting, running dogs, guns, bows, rifles, shotguns etc.
Saying that hunting deer with dogs is in-humaine makes me wonder just how humaine other means of hunting are??
Bear baiting, food plots, feeders, I mean if I was to say that running dogs was in-humaine, how could I ethically use a deer grunt call or lure scent. How humaine is it to fool a dominant buck in to thinking he's about to meet an adversary only to meet his maker.......
Saying that a clean dog run deer kill is in-humaine, makes me wonder how humaine my bow kills are when the deer runs 60-100 yards before laying down to die.
Everyone has thier own hunting practices, just because you don't practice this way doesnt mean it is un-ethical or in-humaine