I have not been on a tour or with a group of any kind in Europe for about 40 years now; I have always planned our own trips with my wife and occasionally with another couple as I like custom-designing my itineraries.
The Gate 1 trip has a lot of great cities, but boy... that's a really rush-rush trip and you'll be left wanting more at almost every one of these cities. It's almost impossible to identify which was my favourite as they're all great in their own unique ways. I'd probably rank these in order: Prague, Krakow, Berlin, Budapest, Warsaw and Vienna. All are great, enjoyable cities. Ask ten people on this forum and you'll get ten different orders.
13 nights... I would cut out one or two of these cities and save them for another trip, if possible. You could easily do Poland on its own or Vienna/Budapest on its own to be combined with other great cities such as Bratislava, Zagreb and Ljubljana. Two years ago, we went on a 23-day trip to Prague, Trencin, Bratislava, Budapest, Vienna, Brno, Telc, Cesky Krumlov, and Plzen, still felt like we needed more time to properly appreciate each of these places.
Have a great time plotting out your trip!