That's my memory of the case.
The Bundys did own grazing rights to the land, as long as they paid the fees. As I understand it, they paid fees from the 50s thru the 90s, at which point Bundy stopped paying. I believe he thought he had acquired vested rights to graze his cattle for free on land he did not and never had owned.
Of course, if he owned it, he would have had to pay taxes on it, etc. Not to mention tie up a spectacular amount of capital even assuming a very low price per acre. The amount of land affected is somewhere between 150,000 and 500,000 acres. At $250/acre that’s $37,500,000 to $75,000,000.
Courts have found that the Bundys started running cattle in the area in the mid-50s, not in the 19th century.
As I said, this is not to say that BLM policies were right and proper, just that Bundy never had much of a leg to stand on from a legal POV. Just because someone is in conflict with the federal government doesn’t mean they’re in the right.