I see nothing wrong with driving this route.... especially if, like you, you have plenty of time. And having a car, especially with a family is going to be so much more flexible than taking trains/plains.
Driving long distances after a TATL flight might indeed not be the best way to go, why not find a decent place around FRA to stay for your first day to familiarize yourself in a less crowded environment.

Just be aware that driving speeds on average will be "somewhat" higher than in the US. Keep looking in your mirror when in the left lane on the German autobahn at 120 km/h, someone might be driving 180+ km/h -). And traffic around the major cities can be a bit hectic as well.

But why you would stay for 3 weeks at a Marriot in Disneyland or 3 weeks in Rome beats me. Having a car is for driving, not for parking it 2 x 3weeks. Take a trip driving around Italy, that's the experience to go for.... Not eating McD at Disney when in France/Italy. Food along the major highways will be just as bad as in the US, no tips required here.