One issue is that hunters hunt differently than wolves and other natural predators. Predators go after the old, the sick, the weak, the young. Hunters almost exclusively go after the health. Not only is that changing the distribution of the gene pool, it also isn't effective population control. We'd need to force hunters to go after a bunch of fawns instead.
There can also be unintended consequences to human culling, besides long-term genetic changes. There's one example of when they started culling a elephants in Kruger National Park. The Park had too many elephants and so they had to kill some of the herd. However, they tended to kill only adults, leaving a lot of juveniles around. The problem is that, once juvenile males reach adulthood, they leave the family herd and hang out with an older male. The older male teaches the younger males about surviving in the wild. And, importantly, self-control. Male elephants periodically go through an extreme spike of testosterone (called musth) that can cause very aggressive behavior. The young males learn from the older male to control themselves. If there aren't any older males around, they don't get this knowledge, and will be prone to violence. After that culling, there was a large spike in incidents of these rogue elephants attacking villages and rhinoceroses. In short, culling can change the behavior of the animal population immediately and for the worse. Not sure how relevant that is to white-tailed deer, but it's something to consider.
There can also be unintended consequences to human culling, besides long-term genetic changes. There's one example of when they started culling a elephants in Kruger National Park. The Park had too many elephants and so they had to kill some of the herd. However, they tended to kill only adults, leaving a lot of juveniles around. The problem is that, once juvenile males reach adulthood, they leave the family herd and hang out with an older male. The older male teaches the younger males about surviving in the wild. And, importantly, self-control. Male elephants periodically go through an extreme spike of testosterone (called musth) that can cause very aggressive behavior. The young males learn from the older male to control themselves. If there aren't any older males around, they don't get this knowledge, and will be prone to violence. After that culling, there was a large spike in incidents of these rogue elephants attacking villages and rhinoceroses. In short, culling can change the behavior of the animal population immediately and for the worse. Not sure how relevant that is to white-tailed deer, but it's something to consider.