I think some other airlines allow redeeming miles for car rentals (either standalone or in combination with other "vacation package" stuff like flights or hotels) without even needing the airline-branded card. But I'm not sure whether the redemption rates are worthwhile if you have other uses for the miles.

Also, although this isn't exactly redemption of points in my example: Through "AA Vacations" (ie, packages) I once booked a prepaid combination of a roundtrip flight LAX-DFW and an Avis rental in DFW for one price which was significantly lower than the flight and (insurance declined) rental separately through any normal channels, and included full insurance on the rental to boot.

So while Avis free weekends only cover the base price and maybe taxes, some miles redemptions for rental cars may cover everything including full insurance.