Managers role is to manage people, but that doesn't mean to impose limits and micro-managing their lifes etc.

My ideal EM in structure #3 is responsible for direction of the give technology or set of major applications, or projects. They are directly responsible for well-being of their reports, making sure people across different workstreams don't work on silos (which one of the major issues in software teams), carrer path, growth, hiring, etc... I expect EMs to be expert in their tech field. (Usually). They are leaders, not just managers, that's the most important part of their role.

Workstream teams manage themselves, with PMs help to set the goals, communication with stakeholders, etc, as you mentioned earlier.