Purchases/lotteries/whatever allow one entry per natural person, using an ID. You have to indicate ownership of that ID to take up the entry ticket. Then you have to show the ID on entry. Typically such places will buy back tickets and you get a refund if they sell.
You can use 'receipt of state assistance' (in the UK this is called 'benefits') as a marker for poverty [definitely not perfect!] and only allow access to, for example, a low cost lottery on entry for such poor people.
It's not perfect, but is that not at least good?
Here in Germany, the Königssee waterfall had to be completely closed for at least five years to give nature a chance to rebuild, after Instagram users caused enormous devastation [1] - trash, fires, noise scaring wildlife including some species of rare, protected birds, 3km worth of trails through the bushes, erosion caused by the trails... and it's by far not the only place.
[1] https://www.ksta.de/panorama/wasserfall-hotspot-am-koenigsse...
I'm not sure this is true. The United States has a huge amount of wilderness and federal land. The problem is that most people want to go to a small number of very popular places.
Allowing as many people as possible to have a nice time in nature, whether by providing facilities, informing potential visitors of their options, managing access to popular or vulnerable sites, etc, is exactly what the Bureau of Land Management are supposed to do. The fact that there is a single agency in control means that it's not really an example of the tragedy of the commons, either.
That is because they are the prettiest and most desirable destinations. This is unsurprising. People want to see Yosemite, not some barren nothingness out near Ridgecrest, CA or Kingman, AZ.
Just because land is owned by the USFS or BLM and has public access doesn't mean it has any real recreational or attractive value that would draw visitors.
The counterparts to this are that a) many people would rather go somewhere slightly less attractive that isn't overcrowded and b) lots of people are attracted, at least in part, by going somewhere novel.
The fact that there is a certain challenge involved in getting as many people as possible (but no more) to the prettiest sites while finding less pretty but still enjoyable alternatives to those who can't make it is exactly why land management and tourism agencies exist.
So there's two options - limit access to those who can afford it (e.g. by raising serious tourism taxes), or do it in a lottery. Both have obvious downsides: making it expensive is direct discrimination against the poor, and lotteries risk families not being able to go because not everyone gets a ticket or, as we're seeing here, people gaming the system to bypass the lottery or a black market run by scalpers.
There is no good solution.