There are literally millions of places you can camp and recreated for free on public lands in the United States. When the fees do exist, they're for particularly popular areas (National Parks) or cost-intensive uses (campgrounds where you can plug in your RV or take a hot shower).
Campground fees of between $5-30/night are hardly making public lands a "playground for the rich." Same for the $80 (cheaper for seniors, and free for the disabled or veterans) annual pass that'll get you and your family into every National Park into the country, as well as any other federally managed public lands that charge an entry fee.
There are certainly important issues[1] about equity in access to public lands, but we're pretty far from your "plaground for the rich" scenario.
The website is basically a scam because you can't enter if you don't have a pass. But buying a pass doesn't guarantee access. I can't make a reservation either, I have to get there in line (after buying the pass online) and hope I can get early enough before the park is 'full'. This basically means the park is off limits.