I performed these steps:
Installed DNS service (plesk component)
Waited until the installation process was complete, terminating it if necessary
Created a symbolic link from hosting to default dkim private key
Ran Plesk repair (ran now without errors)
After that, outgoing emails were correctly signed with hosting again.
I suspect there are still bugs in Migrator if you don't transfer the DNS component (I assumed that you didn't need external DNS and should actually be the case). Apparently, the mail repair program wants to access its database entries.
In addition, there is the default key issue if you have a different name.
It took me a few hours to figure out how and where the problem lies.