Although I agree that a schedule like this is madness, realistically, what is the difference between seeing 10 things in one city vs one thing in 10 cities; you still see 10 things. Of course, with the latter, if you want to come back then you have to do it all over again seeing a 2nd thing in the same 10 cities. Or maybe after the 10 cities trip, you omit some cities as not interesting and come back to see some of the cities in more detail.
The biggest problem is that you spend so much of your time and money traveling between cities that you probably decide Europe isn't worth the time and expense and never come back. Another problem is that you are constantly packing and unpacking.
However, there is an offering of the Bahn called 7-Tage-Nacht (7 days night), where you use the night train network to travel between cities at night and sightsee during the day. A typical trip would start in Cologne, then go to Copenhagen, Munich, Paris, Hamburg, Brussels, Berlin, and end up back in Cologne. You could do it straight through in a week, or, if you arranged it at the time of booking, spend one or more nights in cities before moving on.
Sound perfect for someone who can sleep on trains and wants a sampler of Europe. The package costs a little as €280 for 7 nights in a couchette-6 or as little as €455 in a 2/3 person compartment.