Yep, astronomers would lose their shit at 1-2km wide solar panels in LEO. GEO makes more sense if you want to stream data to it directly and makes the size less of an issue if it's that far away.
I guess if you already have crazy amounts of solar power you might be able to give it lots of hall thrusters and move it out into a really high orbit after launch. But perhaps that power is always better used to run GPUs so the cost to put it directly into a high orbit is acceptable.
(Not suggesting it's actually going to happen, just theorizing)
All of this (not just SpaceX, other startups and big companies are claiming they will do this) is based on -- predicted but as yet unrealized -- starship launch cost of $100-$200 / kg. That's an LEO target, GEO will still be a multiplier larger.
But honestly, the whole thing about orbital data centers is just a scam, and I'm pretty sure most of the people promoting it realize that. It's just a way to goose stock prices by generating hype.
The fundamental problem with the idea it isn't radiation, or cooling, or launch cost, or latency, anything technical. It's that it doesn't actually do anything. A datacenter in space is *less* useful for serving requests from terrestrial clients than one on the ground, and it costs more. The only reason to launch AI processors into space is if you need them in space, for instance as part of an imagine sattelite or space probe, or even a communication satellite. Just servicing requests from the ground will never make sense.
A big part of this is that people misunderstand the costs with datacenter water and power. People say things like "water and power are the biggests costs of operating a data center" -- This is somewhere between a lie and completely misleading. Water is a negligible cost, and optional: it's used over air cooled chillers because it's incredibly cheap. Electricity is a major fraction of operating costs, but mostly that's because operating data centers is cheap.
Consider a hypothetical data center unit that is similar in size to the planned Starlink V3. That's the upcoming version of the highest volume satellite ever built, and one that's been heavily cost optimized. It is expected to weigh 2000 kg, and it's solar arrays will probably produce around 80 kW of power. A "data center node" would look a bit different with more power going to compute and less to communications, but a lot of it will be the same. If you launch this into a sun synchronous terminator orbit so that it has 100% sunlight, it will produce about 3500 megawatt hours over a 5 year lifetime. At wholesale electricity rates in the US, that's avoiding maybe $150,000 to $200,000 (source:
https://www.eia.gov/electricity/monthly/update/wholesale-markets.php, and also unverified reports of what hyperscalers typically pay). There is simply no way that you can take an equipment rack, redesign it to survive launch and zero g operation, add solar panels, a bunch of communication hardware, navigation thrusters, fuel, etc, and launch it into space for $200,000. And if more generation capacity is needed, it's cheaper, easier, and faster to deploy solar, wind, and battery storage, and natural gas plants on earth than to launch them into space. This is doubly true if you build for your own usage and don't have to wait for grid interconnect permissions, which is definitely lagging in some places.
Of course, not everywhere has cheap land and abundant solar and wind capacity. IF you want to build a data center near new york city, you probably aren't going to have on-site solar power generation. And your power will be more expensive than I quoted above. But if you are willing to build your data center *in space* you can put it in western Texas.
Plus, when you build a data center on earth and your computer fails or becomes functionally obsolete after a few years, you pull it out of the rack, shred the hard drives, and send it for recycling. Then you install a new server in it's place and get to reuse all your infrastructure. With a satellite you just will watch it burn up on reentry.