There are two reasons from landowner/rancher perspectives that I can think of for why they fight to keep the current checkerboard lands and corner crossing regulations (or lack there of) the way they are.
1. The land owner is greedy. Take Elk Mountain Ranch in Carbon county Wy. The famous no trespassing signs posted at corner crossings were posted somewhere earlier in this thread. That guy and previous owners have just tried and succeeded at controlling those public lands as their own. One of the owners even managed to shut down some of the roads that go up the mountain with some bs lawsuit that said the roads interfered with ranch operations or something. The current owner has opened a HMA for cow elk on some of his deeded ground adjacent to the mountain, so he has at least made some what of a gesture to the public. I know of other ranchers in the area that will call the sheriff on every person they see that has corner crossed, and the sheriff and county attorney do prosecute for corner crossing in Carbon co. If someone tried to fight it they may win due to the lack of written regulations, but evidently it hasn't been worth fighting.
2. The more legitimate reason from ranchers stand points is protecting their operations. I know a different ranch manager in the area. He is strongly against hunting b/c he has been shot in the general direction of by people who had corner crossed into interior sections of public ground, had heards of elk come stampeding by him that had been shot at by people who had corner crossed, had cattle shot on very interior sections of public, and has tons of fence to fix every spring from heards of elk that get shot at and then run through fences. He doesn't care to much if people hunt the public sections that have road access. They're usually visible from the road or their vehicle, he knows they won't be deep into the ranch and so on. From his perspective it's much easier to buy into the anti corner crossing view.
Obviously this is a very heated issue from both sides of the argument, and it doesn't seem like there will be any solutions any time soon. I do think that re-deeding some lands would be good for the general public, but a lot of these land owners have deep pockets.