After thinking about what you wrote here I have some conclusion:
If someone is "good manager" that does all those things whether he writes the code or not he is going to be a "good manager" anyway. Explicitly writing code might not be best use of time but hey if person feels like he needs it that is on him.
If someone is "bad manager" that doesn't bother to deal with technical details and wants only to do "important management stuff" and thinks he can manage by proxies like counting story points or counting closed tasks, does not care about HTTP client A or B and learning the system, he is going to be a "bad manager" and will never even care about writing code.
Finally "bad manager" and "good manager" - is hard to tell because "bad manager" can be good for the company or a team as much as "good manager" and depending on many other factors it can be that "down to earth, hands dirty, good manager" might be really bad for the company or a team depending on business context.
If someone is "good manager" that does all those things whether he writes the code or not he is going to be a "good manager" anyway. Explicitly writing code might not be best use of time but hey if person feels like he needs it that is on him.
If someone is "bad manager" that doesn't bother to deal with technical details and wants only to do "important management stuff" and thinks he can manage by proxies like counting story points or counting closed tasks, does not care about HTTP client A or B and learning the system, he is going to be a "bad manager" and will never even care about writing code.
Finally "bad manager" and "good manager" - is hard to tell because "bad manager" can be good for the company or a team as much as "good manager" and depending on many other factors it can be that "down to earth, hands dirty, good manager" might be really bad for the company or a team depending on business context.