Camping restrictions and permits to enter particular areas are actually very rare when you look at the total land area. I believe in every case these are done only to prevent degradation caused by excessive use. By definition, these systems are therefore in place in popular areas, which is why you hear about them way out of proportion to their actual size and numbers.
I have gotten a permit and gone backpacking in the north part of Yosemite National Park. That is a very popular area, and I can see how it would be damaged if everyone could just go there without the numbers being limited. However, there is no limit on adjacent land in the national forest, mostly because the demand isn't there and limiting the numbers not necessary.
There are partial restrictions in the White Mountain National Forest of New Hampshire. These are different due to the different circumstances, patterns of demand, and level of damage overuse can cause. There are no permits and restriction of individuals, and some of the popular trails can get quite crowded. There are restrictions on the size of groups and where you are allowed to camp in the back country. Some of these restrictions move around as damage from overuse shifts. The general rule is to stay 1/4 mile from fixed installations like huts and purpose-built campsites, but also around what would otherwise be popular and overrun areas. For example, you are not allowed to camp near the top of Thoreau Falls. That was done because too may people did and the area became heavily damaged. All that was left was bare ground and large trees. Everything else got trampled. The place looked unreal.
On the other hand, there is no restriction on total numbers of people on the trails. Some of the trails are heavily used and heavily worn, but since that stays local to the trail there is no widespread damage and no attempt to limit numbers. Last summer I did a 10 mile or so day hike that looped over a part of Franconia Ridge that is also the Appalachian Trail. We probably encountered at least 100 people on that hike, and that's not counting the people around Greenleaf Hut, which was on our loop.
There are also many parts of the US that are open to the public where you can hike all day and are unlikely to run into a single person. I have done exactly that in various places in Arizona, for example, in both the desert and the mountains. What makes that possible is the huge amount of public land there. Europeans seem to have a hard time grasping just how vast some of these open spaces are, probably because there is nothing like them in Europe. There are no restictions there because there is no need for them. All the people that want to go there do, and the damage is minimal, so no need to regulate. Take a look at a map of Arizona and New Mexico. Add up the land area of the abutting Coconino, Tonto, Apache, Sitgreaves, and Gila National forests alone, and compare that to the size of a few smaller European countries. Now consider that except for a few towns that are "holes" in the forest, nobody actually lives there premanently.
So in summary, restrictions on camping locations and numbers of people are only done grudgingly as a necessity to protect the resource so that others can enjoy it in perpetuity. Such restrictions are actually quite rare, limited to particularly popular areas. There are vast less popular places where anyone is welcome to hike around and camp with little restriction.
To put the vastness of these areas in perspective, the total land area of the contiguous patch of national forest formed by the Coconino, Apache-Sitgreaves, Gila, and Tonto National Forests is 15.7k square miles. That just about the size of Switzerland, which is 15.9k square miles. That doesn't include other nearby public lands, such as the Prescott, Cibola, and Coronado National Forests, for another 7.3k sqm. There are more smaller national forests and wilderness areas on BLM land scattered about Arizona, and the Grand Canyon National park and lots more north into Utah.
Ryan mentioned Utah. Putting that into a European perspective, it's about the size of Great Britain (Utah 84.9 ksqm, GB 88.7 ksqm) but Utah only has a population of 2.8 Mpeople. In contrast, England alone has 53 Mpeople. And most of Utah's people are concetrated in a metropolitan strip along the eastern edges of the Great Salt Lake and Utah Lake. That leaves very few people living in the large remaining wild areas.
Yellowstone National Park may get 3M visitors per year, but consider it's 1/3 the size of Belgium. The entire national forest system is larger than France, Belgium, and The Netherlands combined. And that doesn't include the national parks, like Yellostone, Yosemite, the Grand Canyon, and many others. 2.7M more people live in London than Utah and Nevada combined, which have a total land area over twice that of Great Britain.
The point is that it's very easy to get a totally wrong impression of wild places in the US because the few well known popular spots are not representative, and by necessity of their popularity must be managed differently.
Europeans also don't seem to get the vastness of the place until they've been here. Spend a day driving accross Nevada on Route 50 and realize it takes more than one tank of gas to get between two places anyone would call a city. Or spend some time in a wilderness area in the Arizona desert and realize you haven't seen a single person in the two days since you left the last paved road. There are many wild, beautiful, and vast places that are open to anyone who wants to come. In fact, that is the case for most of the public lands in the United States. In general, wild areas here are more wild, less crowded, and less restricted than those in Europe. There are also a great many more of them. Come and visit some time.