Picking a metric that reacts to changes quickly is neither necessary nor sufficient, but it certainly helps reduce the error on your calculation. You need to know how far off your set point you are and so you need as accurate a measurement as possible.

Imagine a cruise control for a car where the speedometer had a five second delay. You’d still stay at your desired speed on average but it would vary a lot more and require more work to get back to the desired speed. It would have to accelerate harder and brake harder.