Private land in the US is barb wired, often with no access at all. (This is different from private land in Europe, where the norm is that a forest, in private hands would not be barb wired, and there are sometimes easements for recreational use.)
Devil's Rope: http://99percentinvisible.org/episode/devils-rope/
The freedom of BLM land makes it great.
Once these spaces are privatized, there's no going back.
I love national parks, but the restrictions (while necessary for something that popular/unique) remove a bit of the joy from a natural experience.
Even a more permissive national park (Joshua Tree, for example) feels like a relief after something stricter like Zion or Arches. That said, I can understand why they are very careful with those latter two because they are seriously special places.
Aren't there differences between public use of national parks and BLM land? e.g., most national parks have quite strict rules for camping, while the BLM areas are much more flexible and relaxed? Personally, I think it's great that the two options exist and are distinct. I don't want to discount your general point, but more so wonder if ALL of that land should be set aside for purely private or NP use.
(Arches NP has put backcountry camping on hold. In the best areas of Zion NP, there are fixed camp sites that are quickly reserved in advance, or camping is not permitted. Tourist attractions are getting busier and busier with each year, so BLM options for backcountry overflow make sense to me.)
I'd be worried that it's easier/cheaper to privatise these lands than the reverse.