If an engineering manager (presumably one who attained this position because they were or are a good engineer) is going to fix something, they're going to apply best practices because they know what they're doing.
If an engineering manager was hired just because they know how to manage, well then they aren't really an engineering manager, are they? They're just a straight people manager and in my opinion, shouldn't be managing an engineering team. They should be a project manager.
>presumably one who attained this position because they were or are a good engineer
Why do you presume this? Management is a very different skill set than engineering. The best engineers having to become managers to advance their careers is an anti-pattern which most modern tech companies avoid. Managers thus tend to be average engineers who are good at people and process.
I refuse to believe that managers shouldn't code. Actual managerial tasks do not take up that much time. Like I said in a previous comment, in my experience (as a manager), managerial tasks take up maybe 20% of my time per week. What's going on with the other 80%? I'd be bored out of my mind if I wasn't coding as much as I currently am.