Thank you very much for the information. Yes, the thing with the board is definitely a good idea, but probably not quite optimal in terms of appearance.
Update: It became option 1: Simply bought such a 100 euro shower system on Amazon and installed it. It works better than I expected for the price. Interestingly, two holes were drilled into the tiles in the old shower system (see picture in the initial post). I would have thought that there was only one hole in the grout if you already place it on the grout like that...
