I just tried to boot with just the lid of the NESPi case removed, where the SATA-to-USB adapter is located, which was of course plugged into one of the Pi's USB 3.0 ports.
Now it boots without problems!
Since these problems started after I performed an update of RetroPie, I suspect an EEPROM update of the Pi made the SATA-to-USB adapter incompatible. I never would have suspected the case lid as it has worked without problems in the past.
At least we finally found the problem. I'll now try to fix this by updating the firmware of the adapter in the case lid and report back whether it worked.
Thanks for your help, mitu! :-)