The predator discrepancy makes sense in theory but deer have plenty of predators as well: Coyotes and poachers mainly. Unethical hunters in a solid third place.

Probably the turkey hunters as mentioned above, have been conditioned due to extremely low turkey populations in the past and have not balked at the idea of shooting gobblers only.

Also, they are very different critters with very different environmental impacts if they get overpopulated.