Installed and working perfectly, so I’m answering my own question
This only sends system logs (Settings ->Logs) such as Core / Supervisor / Host / DNS etc to the remote syslog
If you also want to log events within Home Assistant like motion detected, etc (things from (Logbook) on the main navigation panel) I recommend remote_syslog by thebytestuff (GitHub - TheByteStuff/RemoteSyslog_Service: Home Assistant Custom Component - send Syslog message to remote server.) available in HACS as a custom repo. It makes a service available that you can call in automations to log home assistant logbook events. This means you have to write automations for things you want to log, but that seems reasonable to me.
The syslog add on here and remote_syslog seem to play nicely together, I’m running both.
Many thanks mib1185 and TheByteStuff!!!