Starting mongod
manually with the command /usr/local/opt/mongodb/bin/mongod --config /usr/local/etc/mongod.conf
showed that something was wrong with the log file:
FileNotOpen Failed to open "/usr/local/var/log/mongodb/mongo.log"
As it turned out, the owner of the file was wrong. Changing it to myuser
solved the problem. See comments below the question for details.