Well, I got serious and found the solution myself.
The fault was a messed-up as-built APIM. I studied all the dependencies in the cells, and they didn’t match my equipment version. So I flashed the stock as-built dealer version.
Everything works!
And one more thing. All DTC errors are gone.
The only remaining problem is the parking sensor sounds. I’ll check that tomorrow. The cells in 7D0-08-01: xxxx-xx**-xxxx are probably responsible. In the Cyanelabs database, the description mentions other functions, but a small program called “Ford APIM Sync 3/4 As-Built decoder tool” explains that the settings include “Front/Rear Park Aid Adjustable Chimes.”
The moral of this is that when buying a used vehicle, you should first check what the previous owner did to it. 