That's called an on-river dam. Or leveraging an existing lake as the bottom reservoir as is done in multiple places around the world.

To be clear, some deployments of pumped hydro do fill reservoirs slowly with rain instead of pumping water over from a nearby river or lake.

But the point of off river, closed loop, pumped hydro is small reservoirs a long way apart vertically to minimize water requirements and maximize energy. The surface areas are small, and they aren't on water catchment areas intentionally to reduce environmental impacts.