I spent most of my life surveying public lands and I can tell you that the government does not survey land on the request of private citizens. The government surveys the land to facilitate their own land management projects. This is mainly due to time and money, things the FS and BLM are severely lacking in. But also, if they made special consideration to one person or group then they would have to make the same types of considerations to every person or group, and it would become a big mess. One way to get the government to do the survey would be to do a bunch of damage to the public land. that way they will spend the time and money to do the survey in efforts to prosecute you.
There is no reason that a person or a group of people couldn't hire a surveyor to locate and mark corners. But surveying tends to be expensive. Maybe not as expensive in wide open country like eastern Montana or Wyoming, where a guy could just drive around on an ATV with his RTK GPS equipment, as it is in the steep timber covered country around here, but it still costs. In most states surveyors have what is called " Rights to trespass" so landowners can't keep them of their land if trespassing is needed to complete the survey.
All that being said, there really isn't a need to survey. The vast majority of the corners are not missing, just very inconspicuous due to age. Anyone can find them if they learn how to read corner notes, survey notes and plats and get some practice. All of the notes and plats can be obtained from the county surveyor or the State BLM office and the BLMs GCDB will get you close with GPS coordinates. Corner searching was by far my favorite part of the job. It's like a treasure hunt with the survey notes being the treasure map. I was proud to have developed the reputation as a "Corner Hound" because I got pretty darn good at finding that treasure. After I left the BLM and went to the FS, the BLM guys would even ask my supervisor to lend me out to them whenever they were having trouble finding a corner that they really wanted to find. I was always happy to help.
So, spending time in the summer out in nature searching for corners would be a great new hobby. Even more fun than shed hunting, and a lot more beneficial. It's another great excuse to get out there and it could be combined with shed hunting, mushroom hunting, scouting or even bird watching. When you find a corner that was previously lost it is really exhilarating. If the corner needs re-monumentation the county surveyor may have it in their budget to go out and monument it. I don't know if they still do but it used to be that the Fed. had a fund to pay counties for re-monumenting corners.
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