Why would anybody in their right mind go to Italy from Stuttgart, when there's so much to see in and around Stuttgart?

Baiersbronn was already named, Stuttgart has wonderful museums, incl. the Mercedes museum or the Porsche museum in case you're a car geek. The Württemberg wine region is near and around Stuttgart, so a winery tour is also an option. Just look at https://wikitravel.org/en/Stuttgart to find out more. It shouldn't be that hard to kill three days in and around STR.