> a lot of engineers I know who moved into engineering management are writing code again
They should be managing instead. Not to say that they can't code their own tools, but the statement sounds like a construction supervisor nailing studs or welding steel bars. Can work for a small team, but that's not your primary job.
I've been an engineering manager and it's a lot easier to make useful decisions that your team find credible if you can keep your toes in the water just a little bit.
My golden rule is to stay out of the critical path of shipping a user-facing feature: if a product misses a deadline because the engineering manager slipped on their coding commitments, that's bad.
The trick is to use your minimal coding time for things that are outside of that critical path: internal tools, prototypes, helping review code to get people unstuck, that kind of thing.
They should be managing instead. Not to say that they can't code their own tools, but the statement sounds like a construction supervisor nailing studs or welding steel bars. Can work for a small team, but that's not your primary job.