Responsibilities vary. Most schools in our state your AD is a paper pusher. He or she shows up to games to make sure officials get paid and the dressing room is unlocked for away teams. Pay a couple thousand.
Places that care about kids and sports, the AD is the head administrator of all sports, but honestly it is not like that in most public schools. Most schools the AD does not even make personnel decisions. Compliance, Gameday administrating, making sure you have ticket takers and other workers.
Our schools all across the state are filled with teachers and admin who leave as soon as the dismissal bell ends. Teachers who do not coach, sponsor clubs, or even show up to support kids at school plays, games, etc. Same with a lot of AD's. It is not hard to identify which schools have good AD's and which ones do not.