If the designers have half a friggin' brain, that would be built into it... some sort of traffic management (datacenter 1 is busy, redirect to datacenter 2)... kinda like BOINC back in the day.
The datacenter doesn't "choose to go idle", there's no tasks waiting to be crunched, so it just sits idle waiting for something to come in... network/system management _should_ handle the distribution of tasks between all the datacenters... "the datacenter in New Delhi isn't busy, so we'll aim some traffic it's way" (of course, that's only _if_ they set everything up right... one rack-mount server shouldn't be the only one stuck crunching Pi to 20 gajillion decimals for ten hours... send half the task to one, and the other half to another machine, and it takes half the time.
"What's to stop people from submitting the same thing to various AI's": you know as well as I that all this "AI" nonsense will be monetized in short order... you might be able to submit a query for free, but it'll get processed when some server in the football field goes idle, and it'll email you the result. You can get a faster result... just pick from one of 5 dozen tiers of paid subscriptions... the more you paid, the faster the LLM-AI will respond.