Hydro doesn't need pumping to provide load balancing. Just build more turbines [1] than you need for average load - then you can flow extra water overnight and less water during the day while keeping the average the same. This is then equal to the superposition of a traditional and PSH dam, but all in one footprint.

[1] Yes, extra turbines are expensive. But you'd need to build them anyways for a PSH facility.