Oh, absolutely - Mostly I was speaking out of experience of the Gifford Pinchot, Mount Hood, and willamette national forests. I also glossed over the really complex issue of ownership of public lands - By various definitions, public land can be owned by logging companies, the BLM, the various state forest systems, the national monument system, the national forest system, and, most pertinently, the national park system.
Back when the need for protecting land for public use was a new and rare idea, there were two schools of thought about how to do it. Some, like Mr. Gifford Pinchot (Which my local national forest is named after) saw the land as a resource to be managed. Others saw the need to conserve what was there, protecting it from being harvested. The differences split out into two separate systems - The national forest system, and the national park system.
The national forest system allows mining claims - Find gold or other rare minerals on its land, file the appropriate paperwork, and you can dig a mine halfway to china without anyone batting an eyelash. It also does timber sales, selling the trees on sections of land to whoever will buy them. Permits are available to harvest any goods on them for commercial purposes - Lots of huckleberry pickers around here, for example.
The national park system, which includes my local Olympic national park and your Denali, is completely different. No mining claims are allowed, no timber sales are done, and the parks are usually protected from their visitors by a labyrinth of bureaucracy. Seriously, I am not sure I'm smart enough to successfully visit the Olympic national park.
The other methods of land ownership are also worth discussing, while I've mentioned them.
Most logging company lands allows day use on them, provided you don't get in the way of any active logging operations. That makes them a public space as far as anyone in a logging town is concerned - Hunting, fishing, hikes, etc. etc.
The BLM has a similar idea of land management to the national forest system, but I didn't mention it in my comment since, while it owns large chunks of the eastern Washington/Oregon deserts, it owns virtually none of the forests. Not sure why that is.
The state park systems can either allow logging or disallow. The Clatsop state forest in Oregon, for example, sells timber, while the Lewis and Clark state park in Washington does not. That state park is where I saw my first giant - It's less than a square mile in size, though. You'll wander through and see all it has in a few hours. It's surrounded on several sides by logging clearcuts too- How's that for irony?
The national monument system has a very similar outlook to the national park system - Extremely protective and very bureaucratic. It doesn't usually protect large swathes of land, but the area around Mt. Saint Helens has been under their jurisdiction since the volcano blew. It'll be a hundred years yet before there's any kind of forest there again, though, and when that happens it may lose the special status that made it a monument in the first place.