I grew up hunting elk, pronghorn, mule deer and black bear on public land in CO. So far I have tagged 98 deer, 25 elk, 17 pronghorns and 2 black bear as well as other game. With the exception of about a dozen deer all my game was taken off public land. I even took a whitetail buck off public land here in TX last year.
There is no reason to be afraid to hunt public land.