Road Trip Thru Germany

Draw it out on Google Maps or via Michelin and ask again.

Nothing wrong with a nice driving tour, but I suggest you eliminate cities like Frankfurt, Amsterdam and Berlin, where a car is a liability rather than an asset. Being a road, the Romantic Road makes sense by car. So do many other destinations in Germany. So I think it would be wise to research some of those smaller destinations that might be of interest to those who are fond of driving. Besides the Romantic Road, there are other theme routes you might consider...

The German Fairytale Route (die Märchenstraße) would be an infinitely logical choice after flying into FRA airport.

Just to the south of Frankfurt, the German Castle Road (die Burgenstraße) takes you from east to west through many nice places:

https://www.burgenstrasse.de/uk/Home.html

To visit Amsterdam, it could make sense to drive the Fairytale Route, drop the car at the north end somewhere (like Bremen) and take the train into A'dam. There are direct trains between Amsterdam and Berlin that take about 6 hours, comparable to a non-stop driving marathon, but much more relaxing. The Man in Seat 61 can tell you all about it:

https://www.seat61.com/trains-and-routes/amsterdam-to-berlin-by-train.htm

I am currently on my way to Munich from Berlin. We arrived here by train from Frankfurt. I will rent a car for the last two weeks of our travel. I will collect it in Nuremberg and return it to FAM.
I would not recommend a car here in Berlin. The available transportation is excellent. A car might be a bit of a barnacle.
And from past trips, I see no value for a car in Amsterdam...in fact less there than here in Berlin. I drove from Paris to Rheams, Rheams to Dinant, Dinant to Brugge, Brugge to Amsterdam and immediately got rid of the car when we got there.
I am not one of those who caution against driving in Europe, in fact I encourage it, but I would not do as you are planning. There may be a better way to skin that cat.

Not that it matters, BUT Frankfurt to Amsterdam to Berlin to Munich and back to Frankfurt is clockwise.

BTW, there is nothing remarkable about the Romantic Road itself. It's just a two-lane country road clogged with farm equipment, lorries, and tour buses. It's the town along the Road that are of interest. I particularly liked Bad Mergentheim (Deutschordensmuseum), Nördlingen, Donauwörth, and Landsberg, and, of course, there is also Rothenburg odT.

Don't just drive the Road and not visit the towns.

I suggest that you avoid having a rental car in large cities like Amsterdam, Berlin and Munich.

You should do Germany prior to heading to Amsterdam. If you have already booked your flights, after landing in Frankfurt head to Berlin (unless you can change flights and fly directly into Berlin), from Berlin, head down toward Bavaria, consider stopping in Dresden , Nuremberg to Munich. From Munich visit Salzburg, Austria and/or Garmisch/Fussen (do the Zugspitze and the castles).

Head up the Romantic Road through Augsburg, visit Dinkelsbuhl, Donauwurth and Rothenburg ob der Tauber. Continue to Wurzburg.
Schloss Lichtenstein is bit out of the way, since it is between Switzerland and Austria.

Doing the Black Forest (don't miss Strasbourg, France) go up the Rhine valley, visit Heidelberg and Mainz, Rudesheim, Cologne then on to Amsterdam.

Schloss Lichtenstein is on the edge of the Swabian Alb, just south of Tuebingen. Not in the country of LiEchtenstein

While I agree that you have enough time to enjoy the illusion of "seeing Germany", this is a very big and varied country. You may have a pleasanter vacation if you lower your expectations of "seeing everything." We happened to visit Munich on a non-car trip through three Austrian cities, long before we made multiple separate trips to Germany.

My personal take is that a car is useful for seeing multiple destinations in a small area on the same day. That's because most train systems are radial, and your second desired "sight" might require returning to downtown to take a different line. On the other hand, using the car to make big jumps is highly "American" error. When you see a swell tourism sign, it will then take you 15 minutes to drive off the (slow and congested) highway to downtown, find a Parking sign with empty-space counts, and drive to a paid indoor parking garage with narrow aisles and tiny stalls. Then you walk to what you really wanted to see. Ask yourself, "Is that really 'the freedom of the open road?"

To be more concrete (NOT recommending specific visits to you), here is our arrival day plan for a past trip. We did not manage to make all these stops!! We deliberately selected an out-of-town hotel, because we've been to Frankfurt before, and it was not a priority for this trip; FRA->Höchst->Idstein->Bad Camberg->Limburg?->Bad Homburg->Kronberg (hotel).

Absent a list of actual destinations, one basic improvement would be fly into Amsterdam, and train to Aachen or Cologne (both great visits) and rent the car there. You will be really sorry if you take a rental car into Amsterdam!

Edit: Typo only