If you had three weeks to travel in the area circled in red, where would you definitely not miss?

>Should I brush up on my college German before the trip? Or will English be mostly fine?
English will generally be OK in Germany, Austria, and “German”-speaking Switzerland, but you might as well brush up on German if you can actually speak it.
>Should we rent a car and drive? Or is public transportation good enough?
Varies a bit by country, and depends at least a little on how rural vs. urban you want to get, but will say that public transit here in Switzerland is excellent. For most people there’s no need for a car unless you’re specifically interested in enjoying scenic drives, for which there are multiple good options. But there are scenic train trips as well.

Cars are good for very loose schedules and out-of-the-way destinations, but in or between major towns and cities they’re not worth it.

If you decide to come to Switzerland, most people will suggest avoiding, or at least not focusing on, the cities; there’s an argument to be made for this, as scenic mountains may be more impressive than extremely expensive, not necessarily all that interesting tiny little urban centers. I don’t think there’s much reason for a tourist to go to either Zürich or Geneva, although neither is a bad place; they’re just rather dull and extremely pricey compared to other European cities. Bern’s OK for a day, with an attractive little old center and a pretty nice natural setting, as well as being a place to stop off en route to nearby Bernese Oberland mountain vistas; Luzern is likewise kind of nice and has some hikes/scenery just outside of town. Lausanne, on your list, is arguably nicer than Geneva for a tourist if you want some French-speaking Switzerland.

Not skip but you have selected a large area across probably a dozen countries on a modern map. You need to narrow it down a lot and give some idea of what kind of reasons you would pick to visit one place over another. Regardless, definitely come up with a plan that optimises the travelling between places according to whatever your aim is. There is a flixbus overnight from Venice to Lucerne which can save you a nights hotel cost in Switzerland. The flixbus from Zurich to Munich is several times cheaper than the train and almost the same duration. There is a transfer tour between Munich and Prague which is a full day but stops off in Regensburg, Plzen and Karlstejn, if your goal is to see as much as possible in a short duration...

Yeah the more I look at it, the more I think skipping Switzerland might be the right call.

Not skip but you have selected a large area across probably a dozen countries on a modern map. You need to narrow it down a lot and give some idea of what kind of reasons you would pick to visit one place over another. Regardless, definitely come up with a plan that optimises the travelling between places according to whatever your aim is. There is a flixbus overnight from Venice to Lucerne which can save you a nights hotel cost in Switzerland. The flixbus from Zurich to Munich is several times cheaper than the train and almost the same duration. There is a transfer tour between Munich and Prague which is a full day but stops off in Regensburg, Plzen and Karlstejn, if your goal is to see as much as possible in a short duration...

The transfer bus looks neat, thanks for the recommendation. There's another one from Prague to Vienna, but I'm thinking we may want a full night in Brno. The train from Vienna to Venice runs through all the stops we would want to do in Austria and Italy - Linz, Salzburg, Innsbruck, Trento, Verona.
Plan as of now is to fly into Stuttgart, do a daytrip to Strasbourg and another to Heidelberg, then Munich, then the transfer bus to Prague.