First, I quote from the Eurail website, "Eurail is not a railway company itself. This means that there is no such thing as a Eurail train [my emphasis]." The ICE from Stuttgart (note spelling) to Zürich is operated by the national rail company of Germany, Deutsche Bahn.
The problem with taking an all rail connection from Rothenburg to Stuttgart is that Stuttgart is basically SW of Rothenburg and you leave Rothenburg and go NE to Steinach, which is on a line that runs NW to SE, from Würzburg to Munich, roughly perpendicular to where you want to go. It is really roundabout, and takes extra time, by rail to Stuttgart.
The Bahn schedule shows a bus leaving the Rothenburg Bahnhof at 9:08, going to Schrozberg at 9:40, where you can catch at train at 9:59, arriving at Stuttgart at 11:53 after a change in Crailsheim.
If you hurry you can catch the direct ICE to Zürich at 12:04, arriving at 14:48. That makes the whole trip 5h40m, about 2/3 of the 8 hours you thought it would take. Or, you could spend two hours in Stuttgart, have a nice lunch and look around, then go on to Zürich on the 14:04 ICE, also direct.
Those direct ICEs don't stop in Tübingen, but they do stop in Singen and Schaffhausen, which are about an hour from Zürich and supposed to be nice. You could stop there for the night if you wanted to. From there to Zürich might not warrent a day of your pass; P2P tickets might be less.
The bus from Rothenberg to Schrozberg will probably not be included with you railpass, but, nevertheless, the fare should be less than €5 pP (most likely around €3 pP) and worth it for the time savings. You can use your pass from there.