I have been forwarding the syslog from my other RasPI boxes to a remote syslog so that I have a good history and can spot trends should problems arise. I also send my Ubuntu boxes and my tas-ma-tized devices syslog to the same server.
This offloads this usually giant log to somewhere else preventing wear on my sdcards and saves space for other things.
An add-on that would handle this in HASS.io would be awesome.
I believe there is a facility to send your homeassistant log to the syslog, I saw it in there somewhere. Combining these 2 would push all the logs to a spinning device and get them off of the PI completely.
I believe several people have asked for that to happen, this may be the easiest way.