Hydrostatic valves only open when the pressure OUTSIDE and UNDER the pool is considerably greater than the pressure INSIDE the pool. This can only happen when ground water levels are a foot or more higher than pool water levels. So . . . if your soil drains like you say, any hydrostatic valve could not have opened.

Put another way: conditions that would open the valve, would be even more likely to float the pool!

If it leaks MORE when the pump is OFF, than when it is on, it's even more likely you have a cracked underground suction line. In such a line, when the pump is ON, the pressure inside the pipe is reduced, thus reducing the leak.

. . . Still sounds like a popped (floated) pool to me.