Your points about managing (interfacing with the company, translating company’s demands into a reasonable, steady stream of work) are spot on, but I would also note that the expectations for engineering managers are all over the place right now.
In my first engineering management job I was severely chastised for writing too much code, even though I had the time and was choosing non-critical tickets for myself. The company insisted that managers should manage/delegate and developers should develop with little to no overlap. I internalized this as the way management was meant to be.
Later in my career I took another management role and followed the pure delegation strategy. I was later chastised for not writing enough code. They believed managers should only need to spend about half their time managing and the rest should be spent doing IC work. They never communicated this, but assumed it was just how managers knew they were supposed to operate.
Even later in my career I took a “director” role, which was appropriate for my career progression to that point. I was stunned when the VP I reported to informed me that I needed to be spending “at least 2/3” of my time writing code, and that managing shouldn’t take more than an hour or two per day. They then proceeded to book me into 20-25 hours of recurring meetings every week. That company had major problems for other reasons and it didn’t work out, but after that I always made a point to explicitly ask about manager expectations multiple times during an interview.
Different companies have wildly different ideas about what engineering managers should do.