My dark theory is that the goal is to run an AI overlord in space such that it is difficult to counter it from Earth.

If you assume that these people aren't completely stupid, then there is some reason why they want this workload running at great physical distance from all the people down on Earth. It's probably not to protect people on Earth. After all they'll happily deorbit satellites and other junk from orbit and let it rain down on us. And they will happily destroy the environment with all those rocket launches too. Therefore it must be to protect the workload from us.

What is a workload that is something that people would probably want to destroy, and which would also provide enough value to offset the expense to launch and run in space? The only thing that might make sense is a military AI platform. Think something that observes Earth, launches missiles, and controls terrestrial drone armies remotely, with relatively low latency.

It gets built and launched thanks to endless military budget, and once it is up there, running such an AI from space means that effectively the only people who can take it out are nation state level foes who can launch rockets into low earth orbit. And this thing is a satellite, probably part of a network that is watching the Earth all the time. Start building something that looks like a rocket launch site, and the AI will see, then you get hit by a missile or taken out by a drone first before you get a chance to attack the platform.

It sounds like sci-fi, but in the future, if we let it happen, there could absolutely be nearly invulnerable autonomous AI platforms in space overseeing everything, and making decisions, and issuing commands. Of course there could still be a massive solar flare event, or a Kessler syndrome event that releases us all from AI enforced servitude. Anyway, it's a not so fun thought experiment, and let's hope this stays sci-fi, so we can just enjoy a fun Hollywood film about this rather than experiencing it firsthand.