I would say that if land was purchased that completely blocked off access to public land without such easements, somebody screwed up royally with regard to such easements. Not sure if that's what actually happened with the land in question?
edit: actually reading an article linked within the one linked here: "This is a murky legal area, due in part to the failure of Congress to ensure access to landlocked federal public lands". So yeah, someone screwed up royally and it was Congress.
I’m not sure what rules there are if there’s a navigable waterway that goes into the public land through private land, there may be issues if you have to portage around obstacles on the river onto the private land. I would assume that if you can access the navigable waterway from public land, you are allowed to use it to reach the landlocked public land by traveling along the waterway through private land, but I’m not 100% sure.