Interesting. In the Big Tech companies I've worked at, the manager is the people-oriented equivalent of the technical team lead and controls promotions within an org structure. Why do you need one? Generally to free the technical lead's time up to deal with people concerns less. In smaller companies this doesn't matter as there just aren't that many people to deal with but at a larger company, a big technical project can interact with enough different stakeholders that you need a dedicated people manager to deal with them.

Managers are also expected to help their direct reports grow by offering them feedback. Manager feedback and regular performance evaluations eventually lead to raises and promotions which then builds up a manager's experience level. This often becomes a target in itself where managers try to optimize to promote their direct reports at the expense of more holistic growth so they can get a promotion.

Each team gets a manager because the manager controls promotions at the company and maintains employee standards.

In my org we'd call what you do a coordinator. Handles prioritization but is not above the team in hierarchy.