I don't actually think this would be hard to imagine. I've been a huge fan of Space X since it's launch exactly because these types of things do seem feasible because they save so much of money if they are achievable.
A moon base with a secondary launch site, yes. Mining asteroids for precious metals, definitely. I'm not some Luddite.
My only point here is that you can build a data center on the ground trivially easily. Any data center that can exist in space could much more easily exist on the ground... where you can update it and fix things that go wrong. The only issue is politics. I'm entirely happy to be wrong here. If someone can explain the thesis, I'd be happy to get on board.
A fully reusable Starship will drastically change the economics of both initial launch as well as maintenance. I expect it will become more feasible to send vehicles to refuel/repair/replace components and keep satellites flying longer. Especially for orbital compute where there will be relatively few dense orbital planes. SpaceX showed modular servers in their video https://youtu.be/k3Un1TizSNg?si=14-bjxXkiyM6cxpg&t=36
I've heard this often. It's not what happens. More correct to say that engineers (not Musk) got a booster to land in a pre-specified spot. The "chopsticks" aren't waving around to catch anything flying by. The booster comes to them.
That will make it clear it’s not actually a giant Mechazilla robot reaching out to grab the booster using literal chopsticks where ever it happens to come down.