This actually did help me to find the solution. Thank you very much!
For everyone else, here it is:
shell_command:
delete_old_backups: "ssh -i /config/private.pem -p 22222 [email protected] 'find /mnt/data/supervisor/backup -type f -mtime +20 -name '*.tar' | xargs rm'"
- Replace "192.168.XXX.XXX " with the IP of your HassOS server
- Place a working private.pem file with access to your HassOS server in your config folder and give it chmod 600 (see Debugging the Home Assistant Operating System | Home Assistant Developer Docs)
- execute the shown ssh command once manually on the command line of the homeassistant core container to make sure, the public key of the HassOS server is added (see Execute in Home Assistant Container Context - #4 by CentralCommand)
- The +20 in the command defines, that backups older than 20 days should be deleted