Shoveling around money and work is only a small piece of the job.
A good line manager does things like help resolve interpersonal and inter-team issues, helps address the issues causing underperforming team members to underperform so that they can improve instead of be fired, handles firings and layoffs when necessary but only as a last resort, makes sure team member career goals and skills get considered as opportunities arise, shares concerns and updates both up and down the chain, advocates upwards for necessary staffing and worthwhile raises, oversees hiring for the team in collaboration with the recruiter and tech lead, explains downward for applicable constraints and works with the tech lead how to apply them to the tasks at hand, and so on.
These have all been my goals in my line manager jobs. Notice I said nothing about technical matters or project management or driving execution. That’s tech lead stuff, with some oversight from the manager to make sure business needs are met.
People need management just like computers systems do, but the skill set is totally separate. Computers always do what they’re told, even if software bugs sometimes mean you didn’t tell them you think you told them. People have feelings and needs. Very different.
For a team of more than a few members, management is a full-time nontechnical job. For a 2-4 member team, yeah it can be split.
A good middle manager does the same kind of thing as I said a good line manager does, but managing managers and their teams/orgs instead of individuals.
A bad middle manager does what you think a manager does, and/or several other failure modes.