I thought you needed two dams. One higher than the other. You pump water back and forth between the two to generate or capture energy.

Is it that common that dams are already existing in nearby-ish pairs with the sufficient height difference? And that we haven't done this already?

Doing this is good where we can. But it has geographical limitations. Batteries don't so much.

My layman's understanding is that most locations that are suitable for pumped storage already have pumped storage built on them.

Yep, this is the issue. That and land cost. Also pumped hydro is most useful when you need very large capacity storage, whereas for preventing blackouts you need very high capacity fast generating to fix oscillations or to allow more generating capacity to come on line. They are basically acting as decoupling capacitors (except for AC) in this application.

You need two sufficiently large bodies of water close to each other at different elevations. You don't necessarily need two dams - for instance, the Ludington pumped storage plant adjacent to Lake Michigan uses the lake as the lower body.