The SMTP channel was actually missing. Thanks for the hint.
I installed zammad on another machine and recreated the channel using the defaults from there via psql. Now it works.
To the reason:
We migrated from OTRS and deleted groups we didn’t want to use in Zammad. I know, do such modifications at your own risk. I just want to be honest here.
I think we deleted the group that was connected to that channel and the channel with it.