So what? Water used for cooling doesn't disappear.

Historically large data centers used evaporative cooling. So the water disappeared in sense that it's no longer available in a useful form in it's original locations. That's changing with growing use of closed-loop systems.

A lot of people think it does, surprisingly. People memed too hard about "querying ChatGPT drains lakes" and anyone who contested it was deemed a booster, so it stuck.

I thought some cooling setups aren't closed loop?

https://en.wikipedia.org/wiki/Water_cycle

It doesn't mean the water goes back into the same sources it was pulled from, though

no it doesn't but it doesn't go back to where it was taken from either. So it does affect local aquifers.