Our host has lots of good travel tips in his blue menu top left of Desktop view. He does spend a bit too much time on Rail Passes, which are seldom the best bet for 99% of posters on this particular newsboard.
https://www.ricksteves.com/travel-tips/transportation/trains
Do you have to fly home from Stuttgart? I'm sure Lufthansa has lots of flights from Munich. After dozens of European trips, we now (admittedly, East Coast USA) embrace two segment flights. We've had good experiences, and very low costs. (Lufthansa is a United partner, for example.)
Because you are traveling at a peak period, you have to decide whether you want to buy the fastest trains, with likely required seat reservations, or whether you prefer absolute flexibility with the chance of not getting a seat all the way. (No recent personal experience.) The reserved seat trips are often VERY cheap when bought far enough in advance, but of course the tickets can't be changed to another departure for free.
You may find train travel to be relaxing, if you go with it, instead of feeling like you are not the master of your own mighty vehicle! Because you have only mentioned major cities, I would try an entire trip without a car rental. Germany has especially good local trains. But you don't say if your mission is to visit three tiny cities a day from each base-hotel. It's hard to achieve that without a car. But a car in a major European city is quite an annoyance. (Opinion.)