> unfair competition for work
That's a very good indicator of a bloated institution. People have to compete for work instead of pushing it away or avoiding it because they already have their hands full.
But I don't believe there is a general rule that applies here.
Most great managers I had were deeply technical and involved in the nitty gritty of the projects, including coding the very spiky aspects of a project.
Most mediocre managers I had were very focused on relationship building. The kind of manager that would need a hobby project to keep current, instead of being the most knowledgeable person in the room.