Hm. New clean installation of the same system, now on the internal MMC, and it does not work again. I still have no idea why it previously suddenly started working on the SD card (and it still does). Neverending story…
Comparison of the working and non-working installation showed the key difference is dhcpcd.service running on the former one, but not on the latter one. The dhcpcd actually automatically activates and configures the new network device as soon as it appears in the system. If it is not activated on the side of the device, it is not detected by the host either!
For some reason, there seems to be some difference between the images. I do not remember to have activated dhcpcd myself before.