Have you checked with your bank to see if they have a card that works like that? I have a citi American airlines one with bills that are on auto pay to the card and I get rewards. Maybe it depends on how you pay the bills if you get rewards. Plus if you sign up for the American airlines card you get 20,000 miles bonus. Since you don't like to fly you can always do one of their cruises or vacations with the miles.

My American Express give cash back for purchases. different purchases have different %'s. Cash back received as the end of year. We even purchased our motor home and cash back.