First trip to Germany - 9 days

"At this point, I'm thinking Frankfurt to Wurzburg - then take the Romantic Road to Fussen - then to Salzburg - then to Munich. From there, we'd like to drive to Berlin before heading back to Frankfurt. Is this too ambitious?"

For 9 days, yes.

With one exception, the only specific places you mention have 100,000+ inhabitants. Yet you "prefer the smaller, older towns, more interested in the countryside and the people, castles and churches . . . less interest in museums and big cities."

I suggest you do some more research into the sorts of places you are referring to. Have a look at the smaller towns on...

the Burgenstrasse or Castle Road
the Fachwerkstrasse or German Half-timbered House Road
the Bergstrasse (towns are listed at this page in German)
the Deutsche Weinstrasse or German Wine Route

You could also look into the Main River and Neckar River driving routes - there are many nice smaller places there, like Miltenberg and Sommerhausen and Bad Wimpfen and Hirschhorn

Most of these places are within a reasonable distance of FRA airport.

It would make sense to turn in your car after about 4-5 days and to take the train to Berlin for 3-4 days after that.

Well, 9 days is good, but not a lot of time. How much time in Berlin have you planned? Frankfurt is a good base for a lot of places - Heidelberg, the Romantic Rhine. Really, I'd drop the trip to the south. Instead, I'd go to Leipzig on the way to Berlin. Huge war memorial, Bach's grave. I'd do 3 nights in Frankfurt, 2 in Leipzig, 4 in Berlin, or something like that. Salzburg is pretty far from Frankfurt.

Give yourself the time to enjoy unexpected discoveries. On my first trip to Europe I over scheduled (wanting to see it all) . It was tiring and dictated by train schedules and hotel reservations. I just had to promise myself I would return....and I have, 19 times! Each successive trip I cover less territory and pack lighter.
I would whittle some sites off your itinerary...and promise yourself that you will return!

Just strolling through some events and found this if you will be around Wuerzburg Sept 22.
https://en.franken-weinland.de/veranstaltungen/culinary_wine_walk-282622-11862180/

Iphofen is just east of Wuerzburg. A "Picturesque Town" according to my old "Strassen & Reisen" German road atlas.

Rothenburg ob der Tauber is worth an overnight stop. Rick loves it, and yes, I like it too.

Yes, too ambitious. Berlin needs 3 full days - even without visiting only one museum. Car makes no sense here and also not for transport to Frankfurt; better use train (check rail & fly option of your airline).

From Füssen to Salzburg nobody mentioned German Alpine Road so far.

If you want to rent a car have an IDP or any valid driving permission - not driver's license only.

The Romantic Road has several great towns and small cities.https://www.romanticroadgermany.com

The three medieval walled towns of Rothenburg ob der Tauber, Dinkelsbühl and Nördlingen are great. The last two usually have few tourists.

Augsburg is worth a visit, we lived there for four years and loved the over 2000 year old city.

I think it's too ambitious. You won't get to enjoy the places you're in. Flying into and out of Frankfurt will limit you.

Here are a couple ideas:
Fly into Munich
Day 1 Munich (not much fun when you're zapped from travel)
Day 2 Munich
Day 3 Day trip to Salzberg (I might skip it for the sake of enjoying my second day in Rothenberg on the end and not hurrying to the airport.)
Day 4 Fussen
Day 5 Fussen
Day 6 Oberammergau
Day 7 Oberammergau
Day 8 Rothenberg
Day 9 Rothenberg
Fly out of Frankfurt

OR!

Fly into Frankfurt
Day 1 Trier
Day 2 Trier
Day 3 Cologne
Day 4 Cologne
Day 5 Cochem
Day 6 Cochem
Day 7 Bacharach
Day 8 Bacharach/St. Goar
Day 9 Flex for travel
Fly back out of Frankfurt

We travel by train and even though it's been 4 hours JUST to get to Baden-Baden, we've gone each time. It's so, so worth it.

I think you'd probably enjoy the first itinerary most, but it requires you to fly from two different airports.

If you want to really enjoy the trip pick one area (North or South) and enjoy it.

Some other places to consider.

https://www.schloesser.bayern.de/englisch/palace/index.htm