I bet habitat is more of an issue than varmints. If big timber companies and large landowners would spray/burn on a 2 or 3 year rotation, the population would be fine. Not that predator control wouldn't help, and probably tremendously, but pine stands filled w nothing but pine straw and gum trees isn't the best turkey habitat.
I say fix that, not the limit of gobblers which has been known not to affect the population.