The issue isn't that there are few parks for people to visit, the issue is that these parks are unique and highly popular. There are thousands of parks in the United States, there just aren't thousands of grand canyons, thermal geysers, or mountain ranges.
USA, population 300 million
60x people in 29x area
Scale works differently here.
It is not my experience that the USA lacks in this respect.
But if you want to visit the Grand Canyon, Hoover Dam, or Zions, or many of these other places, your last paragraph is spot on. The policy of the State of Nevada has been that you cannot reserve campsites in state parks. I think that just changed within the last year for the two most popular state parks. If you want to camp in the Valley of Fire among the red rocks or petrified forest you now need a reservation. If you want to camp just outside the boundaries, there is no reservation required and no fee. Oh, yeah, and no potable water or restrooms, etc.
Note: the Grand Canyon, Hoover Dam, and Zions are not Nevada State Parks. And you'll have to wait another year now to catch Gold Fever at Buckland Station State Park[0].
0. https://parks.nv.gov/events/gold-fever-12
Edit: Amusingly, the headline of the page at that link proclaims "Nevada State Parks Launches New Reservation System". sigh
I, for instance, live in the Helsinki region in Finland, which is a metropolitan area tightly nestled between two modest national parks, both within a reasonable distance by public transport, and both of which I can simply stroll into without any such restrictions or bureocracy. The US could, if there was willingness for it, strive to offer similar opportunities near its major cities. It's simply a matter of willingness.
The other issue is probably that some of your national parks are so iconic (e.g. yosemite) that they'd probably have too much demand even if the country was filled to the brim with alternatives, but you probably can't do much about that issue.