I found myself the problem…
To all of you with the same problem and, perhaps, to my future self:
Templater will not execute commands on the fly. It needs to be triggered to replace commands with Markdown code.
The solution is to execute Templater during file creation. There is a respective toggle switch in the preferences of Templater.
Another option is to use the the hotkeys for file creation (e.g. daily notes) and Templater. I guess this is what my predecessor meant in Templater not working what's wrong?
Hope this helps. If possible, it would be helpful to merge this topic with the one mentioned above.
Best
Carl