Here's my best guess.

The Hidden People are from the moon, and want to get back to it using the lunar lander...either that, or they just consider the lander or its crash site to be their home. Either way, they can't get back to it due to the insanity machine.

Isaac and two other astronauts were sent to the moon to mount the insanity machine there as a tool for the government to make people crazy. However, something went wrong and it made the other astronauts insane. Isaac sealed them in the airlock and piloted the ship back to earth himself, crashing in Scoggins. The astronauts are somewhat feral and crazy and live at the crash site, defending their insanity machine.

The Hidden People whisper to humans to get them to try to destroy the insanity machine, but whenever anyone approaches it, they are caught and killed by the insane astronauts. All the missing people have been killed by the astronauts. This is why we see the remains of a skier at their campsite. Nelson's long crazy rant to his boss in the diner is more or less correct.

This does not explain why they don't kill Nelson, but instead apparently inject him with a sleeping drug and take him home somehow.

I can also find no explanation for why the agents allow Nelson to screw around with the device while almost naked. They have their guns trained on him and everything. I don't know why they didn't just drag him off like they did the first time.

I have to assume a lot of the incoherence is due to the fact that there had been more Puzzle Agent games planned, but they had to compress the story and finish it here.

Another possibility is that there could be other explanations for the astronauts' behavior that doesn't require them to kill people...it's possible that the skier clothing we see points to the idea that they sometimes dress as skiers and go into town, and could even be that skier we see multiple times waving to Nelson. But if that's the case and they're not crazy killers, then there isn't really a good explanation as to what happened to the missing people.

I have the theory that the missing people tried to fix the machine but failed and went crazy in the process, just like the "Lobster" guy who got lost in the woods and froze to death.

Most of them probably didn't even reach the crash site because of their addiction to complex puzzles. Also, as we see from his personal experience, he was half naked when he went crazy, so perhaps the skier took off their clothes and ran away from the site.

The astronauts just drugged anyone who came close and sent them back to make it look like it was just a weird dream. Or maybe they only did that to Tethers cause he's from the FBI and they knew it would attract too much attention if he goes missing compared to the townspeople (The sheriff doesn't care, but the government might look up what happened to Tethers if he goes missing)