There's no river at the bottom, either. I thought the "desert" part would make that clear, but I guess it had to be said.

Water in this system is part of the capital cost (charging it up when you start), then a minor maintenance cost (replacing evaporation). Otherwise, unlike primary hydro, there is no large constant flow out of the system, and no need to be on a river.

The water lost to evaporation is at least an order of magnitude less than water evaporated from a nuclear plant of the same levelized power.

Wouldn't you need two reservoirs in that case? Otherwise where is the water being drained down to take advantage of the potential energy?

Yes, of course. Off-river pumped hydro involves two reservoirs.

I think the confusion is a couple of comments up you said "There's a plan to put a reservoir for one on top of a mesa in the desert in Arizona" which is really confusing since of course they're planning to put a pair of reservoirs for one in the desert in Arizona. I guess only one is on top of the mesa though.

So at that point why use a river? Couldn't you just use ocean water? It's comparatively "free" and we aren't going to run out anytime soon.

With a high enough plateau near an ocean, that should work.

I think those places are really rare though.