We deal with similar kinds of feedback systems all the time in integrated circuit design (see: Phase Locked Loops), but they're evaluated theoretically, and by extension, I think, more rigorously. I'm sure this is largely due to the fact that what we're controlling has very well-defined, well-understood, and simple behavior, where "PID" controllers are often used in far more complex, multi-part systems like motor/speed control, temperature, etc.
And while I think there's a lot of value in developing that kind of intuition, I've run into a handful of junior engineers at my company who don't deeply understand what's happening in the system and draw incorrect conclusions or make dangerous decisions.
As one example, I saw a presentation showing an oscillation at the output of a tuning system when the sensing element was moved further away from the controller. with changing amplitude based on what the sampling rate was. The engineer had no idea why this was the case - moving the sensor away was introducing a distributed, continuous-time pole into the system, whose location in the discrete time, sampled system was moving with sample rate, and changing the feedback systems phase margin based on its proximity to the other discrete poles/zeros from the PID. Even worse, he didn't realize that in the extreme, the system could go completely unstable with in bounded oscillation!