Suggestions for a Base City in Europe

Rome is one option. Within a few hours for weekend trips you have Florence, many small Tuscan and Umbrian towns, Venice, Naples, the Amalfi coast...many of these aren’t day trips, but neither were your destinations from Madrid so this could be similar.

Bologna is a wonderful university city with excellent rail connections. We have considered doing what you plan.
Prague is a nice city to visit but I do not especially like Central Europe, the food especially.
Lyon France is another possibility.
We also used Nice as a base one time, France and Italy together! The beautiful Piemonte region is nearby.

A cruise up the Rhine from Amsterdam to Basel would set you up for more time in Switzerland.

If "robust travel hub" means a city with lots of connections to good destinations, then Bologna would be a good choice. So would Paris, especially if you're willing to go for some overnight (or longer) trips. Prague could be a good hub for places like Vienna and Munich, as well as Czech destinations like Cesky Krumlov.

Prague is great if it's anything like it was when I was there in 2003. Depending on where you want to travel, how about Munich, Vienna, Venice, Milan, Rome, Paris, or Amsterdam?

From Praha you can also travel up into Poland, via train or bus, to Wrocław or Kraków and beyond. Kłodzko Valley and the Sudeten Mountains to the east also make for fun and untouristy escapades :)

For western Europe, I'd suggest Lyon. It's close to Italy and Switzerland. Easy access to Paris (from where one can go almost anywhere in the world), Germany, Belgium, Austria, Netherlands, as well as London. And it's a great base for seeing much of France. It has high speed rail to lots of destinations and easy connection to the airport from the central city via the RhônExpress tram line.

One caveat: while Paris traffic is worse, Lyon drivers are crazier. Unless you enjoy trying to predict the unpredictable, sticking to mass transit might be best.

I would choose Florence rather than Rome. It's comfortably walkable and closer to places like Venice than is Rome.

I suggest that for a three week period that you consider more than one base. Even using a city like Paris, Rome or London as a base, you need to move on, since you can't see enough without a lot of useless travel.

Example, if gong to Germany, use Munich as a base, then move to the Rhineland and use Heidelberg or Frankfurt as a base.

All excellent suggestions, thank you. Many places we would not have considered. In our planning we tend to return to our travel hub after a few days away, but do not double book hotels. Being creatures of comfort, once we have identified favorite restaurants, plazas, etc. we like to revisit and spend additional time there and finalize plans for our next overnight trip. Hopefully for all of us, we will be able to travel again sooner than later!

Honestly, take a look at Frankfurt. It is right in the middle of Germany, right in the middle of Europe, has one of the top airports and a train system that can be very useful. As a hub, it is perfect. Scenic and historic towns are all quite near, as well as the Rhein and the Taunus mountains. Day trips are easy as well as weekend or week trips, especially using the trains.

Look at Milan as a robust hub, though not as well set up for Prague except by air. Turin, Genoa (should you be interested in an ocean cruise), the French Riviera, Italy as far south as Florence, and as east as Venice and all the fascinating places in between. Tale the Breda Canal boat from Padua to Venice for something different. Take one of the scenic trains over the Alps to Geneva or Lake Lucerne.
It might be easier to have some cities in mind, grab a map, and see if there is a centralcentral point ticking your boxes as a hub.

When I travel through Europe, I seem to end up going through München.

Don't underestimate Bratislava as an alternative to Praha. It's much cheaper, and the locals are much more relaxed than in Prague.