In terms of control theory, the order "delay" is actually a simple filter. What leads to bad outcomes is exactly the described scenario, when the control response too fast for the system, overcorrecting and oscillating. Increasing the order filtering dampens the control response.
I think the source of confusion is that the "response delay divisor" isn't a delay at all. This parameter is the inverse of amplification. If you dampen a signal more then it will oscillate less.
Do people actually do systems design with control theory? I know the authoritative question to possibility - yes, my company does it. But I'm wondering about say, an industry standard formalism to it, such as like how DDIA is household
Somewhat, but for distributed systems I’ve found game theory to be more useful. Which does end up looking like high-order control theory in many cases.
https://mcfunley.com/whom-the-gods-would-destroy-they-first-...
https://en.wikipedia.org/wiki/Bullwhip_effect