Here's part of the problem that seems to be ignored to me.
1976 -- Federal Land Policy and Management Act (FLPMA, the BLM's legislative "charter") repeals homestead laws and establishes policy of retaining public lands in Federal ownership. FLPMA requires that these lands be managed for multiple uses and sustained yield through land-use planning.
Basically before you could own without actually owning. When you surrounded you controlled it.
So in essence, many areas prior to '76 were largely controlled by the surrounding owner. That's why they used the checkerboard practice.