Good idea, but don't destroy the rental car plus on a Saturday there might be a regular race. A day in Basel might also be a good idea and if you really only want to drive: Drive down to Lake Constance, the A81 from Stuttgart to the south is the Autobahn with the least traffic in Germany, you'll often see people from abroad there that test drive Porsches and Mercedes sports cars, going from Stuttgart to the lake and back.
Thanks. I've been to Basel numerous times on business, but I've always found it rather dull. Did I miss something exciting? Also, Google doesn't route me on the A81 Frankfurt->Basel. It seems the 81 is somewhat out of the way. Do you suggest it's worth the diversion?