Thanks for your response. Unfortunately this does not help me further.
I already ran the troubleshooter as well as the log extractor. The log file shows (the event viewer export) shows the same error.
There are not other errors in the Extension log. There is no worker / register / startup log in the extension. Only logs of the extension installation.
The 0.settings file shows the same url as the "RegistrationUrl" in the automation account. So I assume it is connected properly. It also shows the Server as online in Azure Arc as well as the connector checks-in in Azure Automate. The error only occurs if you try to run any runbook.
Everything is running on a fully patched Windows Server 2019 which is well in the required specs (4 Cores, 8 GB RAM)
Disabling the System Assigned Managed Identity on the Automation Account does indeed seem to fix the issue for now.