Here in Arkansas, dogs are legal over most of the state. I have hunted with dogs at least once every year. I think there is a BIG misperception about how hunting with dogs work. Most of the time, the deer does not come by running like all hell is broken loose. The deer will usually keep a good lead on the dogs, and will trot, stop, and listen. Granted, some dogs push the deer harder than others, so you never know, but in my experience the deer very rarely run full speed the entire chase.
Also, most deer killed in front of dogs are not the deer the dogs are chasing, but another deer that has been spooked up. I have used my .30-06 to kill a lot of deer in front of dogs. Only one was flat out running his tail off.