Railroad grants. The theory was that by granting land to railroads instead of money it incentivizes them to build the railroad well. By keeping half the land for the government the government profits from the increased value too. They just didn't think this far ahead.
That explains why it was split up, but not why in a square grid. It sounds like the "corner" issue could have been avoided by using a hexagon pattern, right? That way corners are only shared by sections that also share a side. Then again, I haven't been able to convince my friends that using a hexagon grid for our tabletop RPG games is worth it despite the advantages seeming obvious to me (no need for weird rules about moving diagonally), so I suppose this would have been a hard sell to the late 19th century government as well.
In the 19th century no one would so daft as to think ownership of a random plot of land would give you the right to keep people from traversing it. Someone cutting down your trees, hunting, grazing their cattle or sheep, mining, farming it, yes. Just walking across it? They'd think you were a loon.
And having grown up in Norway, I feel Americans talking of "land of the free" while allowing landowners to block off even their own land is offensive on the face of it.
The ridiculous lengths these hunters had to go to in order to avoid straying onto private land is antithetical to freedom (you still need hunting licenses in Norway; and permission to hunt on private land so there are still potential issues, but worrying about a few meters and the accuracy of GPS to avoid even crossing a tiny little portion of private land is not one of them).
I think it was a sound policy. The problem is that the government never decided to sell the land. If they fold it to individuals, those people can put in easements to access their land but that never happened
The land on which the railroad was built was handled separately. This was land in the vicinity of the proposed railroad, which would increase significantly in value if and only if a functional railroad was completed.
The land wasn't granted to build railroads on, it was granted in payment for the railroads. Towns would naturally spring up near the railroads, and both the government and the railroad companies could then sell the land to people who would live on it, farm it, ranch it, etc.