Alsace is a really lovely and underrated region.
If you don't want to rent a car, Strasbourg is directly on the TGV line from Paris to Stuttgart. I think Strasbroug is criminally underrated on this forum - people tend towards the "cute" rather than "impressive" perhaps. But Strasbourg's entire city center is a UNESCO World Heritage Site and is both cute AND impressive. And you can take the train to smaller villages as well. Just an option if you don't want to drive.
If you do want to drive, you have a wonderful array of villages in front of you in Alsace, the Black Forest, and in Swabia (i.e. the region around Stuttgart) and I see no reason to deal with extra driving when there's so much to see that's literally "on the way" to Stuttgart. The aforementioned Haut-Koeningsburg and "Monkey Mountain" are a great idea for one day.
When it comes to castles, don't neglect Stuttgart - there are two right next to each other in the city center, the oldest of which (1325) houses the state history museum which has been offering free admission for some time and has some special kids areas. It's a well-done exhibit with information in German and English. Other fun activities for families in Stuttgart include the two auto museums (Mercedes-Benz, which is excellent, and Porsche, which is excellent for car fans) and the Wilhema zoo and botanical gardens which is beautiful and has some world-class exhibits, especially the primate section. It's in the suburb of Bad Canstatt, which is often overlooked, which is a shame because unlike Stuttgart it's historical center survived the war mostly intact and it's quite charming. I'm a big fan of the Neckarbiergarten, right on the river, with a small play area for kids. Bad Canstatt also has good transit connections to worthwhile sights, like the auto museums, but also the former Free Imperial City of Esslingen, boasting over 900 historic buildings from 1200-1550, a fortification you can climb to among vineyards, and much more.
Not far outside Stuttgart (but requiring a car) is castle Hohenzollern, which looks exactly what a castle should look like! It's relatively "new" (from the Romantic era of the 19th century) but unlike it's cousin, the more famous Neuschwanstein, the chapel is original and it has a commanding view over the valleys of the Swabian Alb. It's also close to Tuebingen, where you can take a gondola ride in a beautifully preserved old German city that was home to some of Germany's greatest thinkers who attended the old University there.
I could go on but my point is there is more than enough between Alsace and the area around Stuttgart (and I didn't even get into the Black Forest) to keep y'all busy with no need to go out of your way to expensive Switzerland or the like.
