TL;DR: My Y axis was a bit skewed. I fixed this problem by loosening the screw on the Y motor and moving the tooth-wheel just a bit to the right. Then the heated bed started to move way more lightly.
It took me several hours to identify the problem. The hint was that autohoming never really worked on the Y axis.
Don't know why the self-test worked OK but calibration always failed as the firmware thought the endstop has been reached due to increased friction I guess.
