Brussels, Amsterdam ... Zurich

Assuming you meant Classical music, I'd save that for an Eastern Germany trip. Unless you will never have another shot at Amsterdam, I'd prefer to leave it out, since Cologne is so direct from Brussels. Don't you need to schedule day trips to each of Antwerp, Gent, and Brugge? Three nights in Brussels isn't enough. And there are some similarities between the low countries ... Saxophone was invented in Belgium. Come to think of it, did you pick Brussels solely for the Musical Instrument Museum? That's optomistic.

Perhaps you could return the car in Weil am Rhine (Vitra Design Museum and campus), sleep in Basel, and train to Zurich. (Expensive country.) Work the car pickup into your Middle Rhine one way boat ride somehow.

I would not spend that much time in the car, but there is plenty to do on this route. You have to slight either Alsace or the Black Forest. Consider Besancon and Perouges if possible.

Edit: Although Belgium has frequent, cheap, non-reserved local trains, I never got to Dinant (Saxe's home ... ) because it requires two trains. The town attracts non-musical tourists for the view so much that Lonely Planet says that the air smells "... Of stale crisps [i.e. French Fries ...]". Of course everyone orders Frites in Belgium.

Tim - Yes, classical music. He was a military musician for 20 years and now teaches music. Saxophonist, actually, so a pilgrimage to the origin of the instrument is apropos. Was also stationed in Italy and traveled all over Europe but not so much to this particular region. That was pre-me, 25 years ago.
Honestly, the flights were booked with less than a day of consideration so I still have a lot of research to do. I will consider all the suggestions and do my due diligence, maybe dropping Amsterdam, and come back for the finer points.

Keep the ideas coming, please!

I guess you have to throw Bonn into the mix if Beethoven is of interest. His home own and lots of historical sites to see there. You can visit on a day trip from Cologne or stay a day or two. Would not pick up a car until that is out of the way.
Suggest spending a few days in the Mosel Valley, specially if it is over a weekend and there is some sort of wine event going on in one of the towns along the river.
http://www.mosellandtouristik.de/en/wine-cuisine/wine-events/wine-festivals/
Then to Trier, Strasbourg, Colmar. You could cross the river to Freiburg-am-Breisgau and drop the car there, then proceed by train to Zurich.

The hometown of Adolphe Sax, Dinant on the Maas river south of Brussels, is very attractive. I know Sax's house has a historical marker, but I don't know if you can visit. Although a very pretty town, there's not much to do or see there.

Other than Bonn being the birthplace of Beethoven, I'm trying in vain to think of any other connection to classical music along your route. If nothing else, just about every mediumm-to-large city in Germany has a symphony orchestra and an opera company. There's the "Mannheim School" of compositional technique, but in 4 years living near Mannheim, I never once saw any kind of museum, exhibit or performance dedicated to it. And the only composer I've ever heard of in the group is Stamitz.

Suggestions for good home base for the driving part? Köln to Zürich is too far a distance to cover with one home base.

Is taking a car from Germany to France a problem? No, not unless you drop the rental off in France, which will generate a large service charge. Otherwise, you can drive back and forth without restriction. However, if you take the car into Switzerland, you will need to purchase a vignette sticker. Get one at or near the border from a service station.

I'd check the cities on your itinerary for the symphony orchestra schedules and concert schedules, if you plan to attend any concerts - I'm thinking Amsterdam, in particular. The season will be about over in June but some of them may be on tour, giving concerts elsewhere. Also, you could hop a flight from Amsterdam or Brussels to Leipzig, which is connected to Bach, of course, Robert and Clara Schumann and Richard Wagner. The 2017 Leipzig Bach Festival takes place from the 9th to the 18th of June and tickets are already available according to the website.