This assumes underground aquifers are static and they aren’t. For example, the panhandle of Idaho sits atop a huge aquifer. You will find there the Lost River, and if memory serves, the Little Lost River. They are “lost” because the water in them eventually sinks into the aquifer. A lot of it emerges later near Twin Falls, Idaho but it is clear that aquifers are not closed off “water banks” from which there are only withdrawals, not deposits.
I believe it's the other way around. Rivers flow long after runoff ceases because water flows out of the aquifer into the river.
If a river 'sinks into the aquifer', what is really happening is that the water table in the aquifer drops below the riverbed.