There are just many locations for the logs and the seem to have to be handled quite different:
Audit-Logs:
Can be exported and so can be sent to syslog, but: I did not find a way to do that efficient, as "xo-cli get-Records" seems not to be able to use filters like "after/before".
Pool-Log:
Is only visible in the GUI. Did not find a way to send it to syslog, or to export it periodically.
XOA-Log:
/var/log/syslog can be exported, but is using quite nasty "multi-line-logs".
Task-Log:
As Audit-Logs: Can be fetched by: "xo-cli rest get tasks", but did not get filters working, yet.
It would be perfect to have options to send directly to syslog, but a workaround through a periodically running script could be good, too.