I'm having exactly the same problem:
I did a fresh install of openmediavault_5.3.9-amd64.iso and have no access through SSH public key.
My Authentication log looks the same:
"Authentication refused: bad ownership or modes for directory /"
My directory permissions are identical:
drwxrwxr-x 20 root root 4,0K Mai 5 23:38 /
This is definitely not an isolated case.
(For example: Issues with OMV 5)
Chmod 755 / solves the SSH issue, but I'm wondering if there are other directories with wrong permissions. Could be a possible safety issue.