There are several National Parks that allow hunting, it just has to be done in accordance with both federal and state law...an activtity that the state regulates on federal land.
Great Basin National Park Utah
Death Valley National Park Nevada and California
Wrangell St. Elias National Park (the largest National Park in the United States)
just to name a few
But there are lots that don't, like Yellowstone, Yosemite and Glacier.
Once again, it goes to the enabling legislation that created the park. You can graze sheep in Great Basin National Park , but not in Rocky Mountain National Park.
If any state wants to prohibit the carry of a weapon in a specific area, they can. As long as it passes the state judicial review, it would be legal. They can prohibit it in specific buildings, school zones and parking areas of domestic violence shelters. What makes you think they could not add "Everglades National Park" to that list. It is not the Feds doing it, it's the state.
By the way, check with the folks who live in the national parks, like Everglades. You'll find they have state driver's licenses, their cars are registered in the state, and they pay state income tax. All activity conducted on federal property regulated by the state.