Their argument that they are concerned about hunters shooting their little furry dog is such utter BS, this is entirely about landowners wanting to block entry so that the can continue to have exclusive access to those public lands.
And I'm guessing that if you were able to see which landowners have contracts with outfitters to bring hunting & fishing clients onto those checkerboard private/public areas, there would be a high correlation with the landowners who are shouting the loudest about 'corner crossing.'