Which historic city to pick Wurzberg, Bamberg and Heidelberg on the way to Zurich?

The Wurzberg would be an unusual choice, though the scenery in the area is nice. If you mean Würzberg (Wuerzberg), that would still be an unusual choice, but does have some archaeological findings. It would be quite a detour from the shortest route, though.
Should you mean Würzburg (Wuerzburg), that would be a far less unusual choice. As a destination to spend a day, all three are definitely worth a visit. You would not go wrong with either, so I'd choose based on convenience (how far from the route you are planning, time of day you want to arrive). I do find Würzburg a bit less charming overall than the others, but that is a matter of personal taste.
One thing I would definitely not recommend is driving. Is there a reason you want to do that? The train will make a lot more sense for an average traveller on your route (more comfortable, probably cheaper, probably faster).

I'm confused. What is your route? You say Hamburg - (Würzburg (I hope, see above)/Bamberg/Heidelberg) - Zürich but then I see Berlin mentioned?? Is that a typo or is that the route you plan?

If you take the sharp easterly turn then my obvious answer will be different than if you come straight south.

I agree with the previous poster. Is there a reason you want to do so much difficult driving, especially as a solo person in a car?

Have you considered a train? It can do 300 kph which is hugely faster than a car can go - even on a rare stretch of open autobahn, and you have to go the whole country of Germany from North to South, and then turn around and it the other way. And you can look at the scenery instead of the road and go around the cities without having to find very expensive parking....

Thanks for the feedbacks! I did plan my train route before, but with two kids, my friends who lived in Europe before recommended me to switch to car instead. I heard for train you need to reserve earlier with to be there on time, it will be harder with two kids, age 11 and 14. Also tree persons, driving makes more sense.
My friend recommended Heidelberg among three cities. My spelling may not be correct for Wurzberg, since don't know how to try the symbols.
The trip was planning to visit some colleges that focus on design to show my son. Also as an architect, I like to visit some famous building include bath house at Veil and others along the way. Milan because my younger son loves Italy food. I may cut a few cities, make it less stressful. But I do a lot of road trip in US. I don't mind driving.

I lived in Bamberg for 3 years and would highly recommend it for a visit.

I also think this is an absurd amount of driving and parking. Each city visit will start with 30-40 minutes of highway exit routing and garage navigating.

Do you mean contemporary "design", 20th Century design, or earlier? Do you mean studio art, industrial design, or architecture? It's hard to give good advice with what we have.

Thank! I will consider Bamberg. Since this is the first time I visit Europe after 15 years, I try to cover more cities. Personally I would like to visit some famous modern architecture, also want to tour some school with industry design program for my son. But I contacted school most of them are not open yet. I have to visit Perter Zumthor's thermal bath in Vail, 2 hrs from Zurich. Then visit go to Milan, Paris, then universities at Holland. My original plan below, maybe it is a bit aggressive, need to cut down a bit.

Arrive 8/4th Hamburg -Berlin- 8/7 (4 days)
Hamburg-Heidelberg/or Bamberg(5hrs 30m mins) (Day 1) 8/8
Heidelberg/or Bamberg-Zurich ( 4 hrs 30 mins) (day 2) 8/9
Zurich visit city and university (day 3)8/10
Zurich- vals bath house (2 hr 4 mins) (day 4) 8/11
Vals to Milan (3.5hrs-4.5hrs) 8/12-8/14
Milan-2 days
Milan to Lausanne-4 hrs 8/15
Lausanne (1 day visit university) 8/16
Lausanne-Paris-6 hrs- 8/17
Paris-3 days 8/18-8/20
Paris-Rotterdam-4hr 40 mins (visit Tjerk’s son’s office) 8/21
Rotterdam- Delft University - 21 mins (visit university Delft) 8/22
Rotterdam- Eindhoven Central-1hr 20 mins (visit university Eindhoven) 8/23
Rotterdam- Amsterdam -1hr 8/24
Amsterdam - Hamburg (5hrs 2 mins) 8/25
Hamburg 8/26
Fly back Hamburg to seattle -8/27

My pick for a summer afternoon/evening would be Heidelberg. Visit the castle of course, pedestrian zone and stroll along the Neckar. Have a nice dinner in the Altstadt. Bamberg and Würzberg for me need more time. Heidelberg is the famous jewel and for a short visit better. For an overnight try Hotel Anlage. They have limited parking for a fee. Make reservations soon. Have breakfast at a local café instead of the hotel before you leave. Not sure any of these cities and modern architecture go together too well?

Würzburg is one of our favorite towns in Germany but to get the benefit of the town in my opinion you need to visit the residence and cross the river to walk up to the castle. It takes a couple days in my estimation.

I like Heidelberg but a long walk up to the castle is also time consuming.

Bamberg's old city core is quite accessible and pleasing for a day trip. I would select it. The cathedral and town hall are the main sights and within easy walking distance of one another.

I would not prefer to drive into any of them. We lived in Germany for several years and had our own car but tended to avoid cities in favor of very small towns. Trains are great between cities.