Solved

The dishwasher is up and running again, yeah! Thanks to all who believed in my skills, very much appreciated. Yet, I have to disappoint you (after having disappointed myself before). I had to buy a new heat exchanger (in German called Wassertasche).
We tried several different means to degrease, loosen, dissolve, get rid of the stuff in the light brownish canal you see running from the top right, turning left and later right again, ending in some kind of pump. Knocking and drumming also didn't help.
So we swallowed the bitter pill and ordered a new heat exchanger. Came today. Unpacked, installed, plugs set, hoses attached, outlet covers fastened. Test run of 15 minutes and see: ran like a charm.

In my entry you see the old exchanger at the left and the new one at the right. We will hopefully remember to clean it regularly, some way or another, before in 14-15 years we face the same issue. But between you and me: the person who developed this exchanger never ever had any thought about cleaning it. There is no way you clean that very canal, too many curves to go any deeper than the first 1-2 centimetres.