You've got it backwards. It's not that you're exempt because you're not paid overtime; you're not paid overtime because you're exempt.

That's what an exempt employee is exempt FROM - being paid overtime.

If you have 20 direct reports I think it's a pretty good bet that you are legitimately exempt whether you have hiring and firing power or not. And if you are an exempt employee, then there are no circumstances whatsoever under which you are due overtime under the law - even if you were to work 168 hours a week.