I'm going to throw in with the dog hunters. If the state says it's legal, then it's legal and it's also fair chase. Having said that, I'll immediately add that I have not hunted behind dogs, and I probably won't. It's a matter of taste, and I am comfortable with my own taste in blood sport not to go knocking someone else's. I have known guys who hunted behind dogs all their lives, and considered themselves consummate deer hunters. I am not going to argue with them. I just finished Meshack Browning's autobiography as well-- now there was a deer hunter! Meshack hunted with dogs, but he also liked to leap on a buck's back, grab the antler and hang on with one hand while plunging his knife into the buck's heart with the other. Again, not my taste, but one heck of a hunter.
It is important, however, to understand one thing about hunting behind dogs for any kind of game: it is extremely consumptive. Specifically, it takes a LOT of acreage to support it, and it is largely indescriminate of property boundaries. Dogs just simply do not know about property lines and they are exceedingly noisey. A couple of hunters and a pack of dogs can really queer up a large piece of real estate, whether the game is coon, rabbit, deer, bear, or boar. With the world getting smaller and more and more private land disappearing to development and owners who do not like hunting, this is important. On the basis of that alone, I would think some states might reconsider their laws allowing hunting deer with dogs.
In comparison, bow hunting is silent and relatively non-consumptive. The experience of a non-hunter is not affected by bow hunting, or at least is not as effected as with dogs. It can be done on a small plot of land. On the other hand, I do not think bow hunting is essentially more "ethical" than gun hunting-- or dog hunting for that matter. It is a shame folks get to confusing ethics and taste.
By the way, I WOULD think that running dogs on deer was unethical if there was no potential path for escape, like in a fenced enclosure. I watch feral dogs chase deer all the time on my property, and the deer clearly have the upper hand. The deer hardly give it a second thought. Jump shooting them is no different than what's done with rabbits and birds.
Lastly, when you talk about dogs, you also start talking buckshot. I'm not a proponent of buckshot. It is not legal in any of the states in which I've ever hunted, and for what I feel is a very good reason. I'm sure folks can fill me in on the details, but as far as I can tell buckshot is great at short distances similar to bow hunting. However, it quickly loses its lethality, and quickly gives way to wounding.