The duties of the AD probably vary by the background of the AD. He (or she) likely has budget responsibility but this doesn’t mean they have a financial background. Barry Alvarez is (or was) AD at Wisconsin after he retired from coaching. He doesn’t have a finance background, clearly. The AD has a staff of those types. Budget responsibility includes fundraising.
The AD hires/fires head coaches as part of their being Chief Athletics Officer (making up a title).
The goals of the athletic department, per sport, are determined by the AD, President/Chancellor, and the BOT. Some schools are known for certain sports, and focus on those. Duke is an elite basketball power, not in football. Alabama is a football power, and competitive in basketball.
Ideally, you’d want someone with a sports organization background familiar with how universities operate. They may not have played, at least not at a high level. Or maybe they did. Not a requirement, to be honest.
At least this is how I view the job.