I don't understand this. Why is management telling you what to do? In our company, the managers ask/push the teams to improve some metric, but the teams are responsible for the how. The metric is usually high-level, like "reduce the number of bugs reported", not "increase code coverage".

I think having cognitive complexity indicators are great. Many devs don’t think about that so having the visibility reminder is good on average. However it should not be blocking in any manner and it’s silly to drive a rewrite based on it. Engineering leaders should understand the architecture at a visceral level, know good patterns/design, and suggest refactors without static analysis tools.

"having them" is fine, being forced them down your throat is not. I mean like "your hotfix cannot be merged because you touched this piece of ancient code nobody understands and now it's complexity grew beyond the arbitrary threshold, please put all your tickets aside, and make a start refactoring this function because Sonar saud so"

Totally agree! That sounds absurd.