I don't think that is correct. BLM land that is leased for grazing is still open for public use. The leaseholder cannot exclude people from using the leased land for recreational use. They have every right, though, to not allow people to cross their private land in order to access the BLM land. In order for the people to access public lands, they must either have a public access point, like adjacent public land or a county or state road, or have permission to cross private land to access the public land.
I have full sympathy for landowners who do not give permission. I've seen they way people in NM trash the public land, and have no delusions that they would treat private land any differently. But to say that a leaseholder can treat leased public lands as their own private land is absolutely correct.
These, from the BLM website:
"First, BLM land is open to hunting, but you have to have legal access to hunt it. Legal access to most BLM land isn’t a problem. However, some public lands are completely surrounded by private land. If there is not legal access through that private land, such as a county road, you need permission to cross the private land. You are not guaranteed access, even though you are trying to reach public lands. It is your responsibility to know where you are, so use maps and GPS units. It is illegal to post BLM land as private land, but every year a few people give it a try. If you suspect someone has posted public land as private, contact the local BLM office to clarify."
"Unless specifically prohibited, public lands managed by the BLM are open to hunting. Always check with your local BLM office in the region you plan to visit to inquire about closures, restrictions and safety tips before you plan your trip.
It is extremely important to hunt only on lands where it is legally allowed. Private land is open to hunting only if you have the permission of the land owner. If you do not have permission to hunt, you are trespassing and can be prosecuted. Crossing private lands to access public lands is not permitted, unless you first obtain permission from the private landowner. The BLM provides public information, such as brochures and maps, through online and physical public rooms. Find a public room in your state here."
"More than 99% of BLM-administered lands are available for recreational use with no fees."