New information: I was able to use the new Windows app again and tried configuring the buttons there. Unfortunately the Windows app shows the same behavior as the Android app described above.
I then deleted one of the broken switches and re-paired it. This did fix the problem and I could start to configure this switch in Hue Essentials again. I might go for this workaround although it is a PITA to reconfigure 10+ switches.
Be aware that there seem to be other issues as well. While reconfiguring switches I repeatedly run into the error shown below. Every time this happened (about every 4-5 times when saving changes) I had to close and re-open the application. The same configuration change then worked.
It would be very helpful to stabilize this. Fixing the problem with the existing switches would be nice, too, although maybe not worth the effort if nobody else complains.