The living part of the tree is right next to the surface, and it is easily damaged, which is why many climbing areas have rules agains tying onto trees at the top of cliffs, using webbing. Those trees often get no loading at all or if they do, they are loaded to approx 1x the climber's weight, near the ground, with a force directed down. A Hammock is capable of creating a much higher force, because a cantenary force approaches infinity when fully horizontal. It is significantly multiplied even at typical angles. And the force is further multiplied by the lever arm of the tree.
I think the main issues the park has to face is the overuse one you mention. Walking on the ground, for instance is not normally harmful to the ground, but most of the campsites I stayed at looked strip mined.
The other problem is that the rangers may not trust that the average camper is an arborist/engineer/rigger who can make the right decisions about soil conditions, tree types, the different forces that result with instalation, and the size of trees that could bare up under these stresses. The trees rarely grow exactly where you want them in the camps I saw, and I had minor trouble erecting my tarp on some occasions. So campers may push the envelope. Agaist that these sites are often special. Normal looking 2-3" trees on the Niagara escarpment may be 900 years old, so how many risks do you want to take? On the other hand, the stealth sites i saw, were usually more heavily forested, tress were often heavy, and tightly spaced. I wanted to use the Henesey precicely because some of these places were too overgrown to take a tent. The heavy cover was often re-forested, and it was not going to get repeat use, possibly ever. I was looking for sites that might take a hammock, in the case of camp grounds, I often had little no choice what site I would get.
If it was my campsite, your hammock would not get used.
By the way, The force of a 1" nylon strap is not distributed evenly over the inch. I used to climb back in the good old days of swami belts and homemade harnesses, tie that same thing around your waist and see how you like it. You need a load take-off that is hard, and distributes the force evenly across the webbing, knots and line tie-ins bunch the webbing up, and do not load it equaly across the width. I don't imagine any harm comes to the trees from single uses, in most cases, but it is far less benign that letting the poor schmuck who paid 30 buck curl up in the mud wallow they call a tent site.