Earlier quoted context omitted.
100% agree with this. I would say that the other highly likely mistake new managers make is trying to code their way out of problems. It makes sense, right? Previously when you're an IC and a project ran into issues you could just "code harder" and get through it, but that's rarely the right solution when you're a manager and will likely exacerbate the problem itself if you disappear into the trenches trying to code…
Indeed. Purposefully stay off the critical path! Do coding that helps you keep up with what people are talking about. Not coding that is urgent!
Refactors, doc fixes, low priority bug fixes, and tech debt are all fair game for managers to pick up. I do think it's important to keep your hand in.