Can I be the Plan Manager and not take up a paid seat?

Yes!

Plan Managers do not need to occupy an active seat in order to manage Billing and seat assignments. However, Assistant Admins must occupy a seat in order to manage seat assignments.

If you are already a Plan Manager for an active Team Plan and wish to remove yourself from holding a seat, head to your Account page, toggle over to the "Team Members" page, and click the trash can icon to the right of your name. This will remove you from holding a seat, but not from your role as Plan Manager.