Disclaimer: I have never had a hunting dog nor tried to train one. But, in my observation, Labradors seem to be pretty versatile. I've also heard that Wiemeraners (I know I misspelled it) are very versatile. I've always thought a German Wirehair/Chesapeake cross might be wonderful.
I have fallen in love with Chesapeake Bay Retrievers. Extremely loyal, intelligent, and versatile. One of the few hunting dogs recognized in the sport of schutzhund. Mine was heeling off the leash within 6 months of age, and would studiously ignore even aggressive dogs in their own yards or in public spaces. Capable of pulling a small sled or swimming for hours without tiring. At 15 years old, he is past any sort of real work, but I am in line for a new puppy next year.
He did well with upland game, and tracked my siblings when asked...a deer can't be too much different.
Here is some more info: Should I get a Chessie?
As unpopular as this may be, I would say a farm bulldog/hound or bird dog cross would be your best bet. When you break away from lines that have been bred for a specific task, you end up with a dog that is easier to mold to what you want. My family hunts wild hogs with dogs and that's what we use. I've seen people hunt squirrels, rabbits, hogs, and quail with the same dog. They won't be the best at everything, but they are adequate at it all. If you were looking for a dog that would find food for you, that would be the way to go. I would prefer a dog that could find some kind of food all of the time over one that was only able to find one type of food really well.
It's not the breed it's the individual dog. I inherited a cocker spaniel from an uncle that was great for hunting everything you can hunt with a dog including bear. I have noticed that bird dogs often get cross trained for small game, but I have never seen or heard of a hound getting cross trained for birds.
actually the type of dog has everything to do with whether it will hunt or not. the training comes after breed selection. i have a Viszla ( hungarian pointer-type dog ) looks like a small red version of a weimeriner. anyway, hounds are usually for furry game, pointers, flushers and retreivers are for birds, and pit bull types and hound dogs can be used in packs to hunt boars and predators. this isnt a rule, but a general guideline to follow. there are always exceptions. i find that smaller pointers like the brittany and the setters make excellent house pets, as well as great hunters in the field. Viszlas are to hyperactive, and remain puppies for ever it seems like. but after the initial calming down period are great family members.