The only time that one of my dogs ever chased a deer(chased, not ran...big difference) is when she kicked it up while hunting coon. That was my fastest dog, and she never stood a chance of catching that doe. It was halfway across the county before she even knew what was going on.
That is what shock collars are for, to keep your dog off of the trash. Any coon dog that runs anything but a coon, is a trashy dog and should be broke of it with a shock collar. You can't win any hunts with a trashy dog. If you are any kind of coon hunter at all, you would know the difference of when your dog is on a coon or on trash. So, in my opinion, a good coon dog and a good owner combined would never run a deer to begin with.