Not quite. His contention appears to be that he has ancestral title, either to the land in fee simple, or at least to an easement providing for grazing rights (I suspect the latter, given his stated willingness to pay grazing fees to the county). If he actually does have such title, then BLM's denial of grazing rights could constitute a regulatory taking for the purpose of the 5th Amendment's takings clause.
The real question is whether such title is vested in him. He claims it by right of family use for generations, predating the existence of the BLM. It's a weak claim, but not impossible. I'd be very interested to know if there has ever been any formal grant in the past, whether there was a claim staked and recorded, or whether he's just trying to lean on "it was never given to us, but we've always done it." The latter can be a basis for adverse possession, though adverse possession is almost never effective against a sovereign (government) landowner.
Based on very limited information, I think he's probably a crank, but this isn't necessarily the same as an illegal moving where he knew he didn't belong and then demanding normalization; Mr. Cliven--or his ancestors--appear to have been there for quite some time. The details matter.
I'm just stirring the pot. I don't care about a turtle or some old guy fussing bout grazing cattle on govt land. OSA has been cranky as of late so I'd just toss something out there to add to the stink.