Trying to interpret your words here - is this a situation where the landowner has to pay the government for the part they own, or pay taxes on it or something, and the checkerboard pattern reduces the payment while giving them effective ownership over the whole area? Maybe the locals are sort of scamming the feds?
It’s a bit indirect. The private land is worth a lot more if they can connect contiguous lots, expressly or implicitly.
It would cause a huge uproar if the fed sold the land, but this allows them to defacto ‘capture’ it as long as no one asserts the right laws.
By stopped the public from
accessing the intervening public land, it defacto gives them ‘ownership’ (as in they can do what they want with it) on the federal public land, but without having to pay for it, pay taxes on it, etc.
These rural areas are typically pretty poor, and the private land owners usually have no issues influencing county and state level politics. The county this is happening in (Carbon County) has a hair under 15k people in it, and a median income of $62k. Wyoming overall has a population of $
570k and most of them live in 1-2 cities.
Dropping $1m in Washington DC will barely make a dent lobbying wise. Spreading the same over the Wyoming and Carbon County gov’t would be… quite powerful.