Your point is correct, but the details are wrong: coal (and nuclear) provide so-called "baseload" power, while natgas and hydro provide "peak" power (this is why natgas plants are called peakers). Coal has a very long spin-up (and spin-down) time. If you've studied process control, the first-order time delay is on the order of hours to days. Natural gas and hydro have time constants of ca. minutes. Moreover to your point, batteries have very short time constants (but a plethora of of other problems).