I've not seen nor read anything that offered 'autoupdates' to Moodle. Now am gonna switch sides of the fence .... maybe, just maybe, Plesk folks have investigated and found that it's really trivial to update/upgrade a Moodle if using git. And they have built into their autoupdate feature the use of git for Moodle. Most of those who have responded have had past experience with Plesk or cPanel ... yes, that's kinda what you wanted, but am not sure that any have used git yet.
So it might be like Windows users who have suffered for years due to 32 bit versions of PHP and un-stable 64 bit versions. Sometimes someone/something makes a discovery and improves. Once saw a quick demo of a product the 'revolutionized backups' ... nice GUI in front of rsync. If one studied rsync a little one could do same without paying the price for this 'revolutionary' program.
Maybe ... just maybe ... Plesk folks have figured out usage of git with what they are calling 'autoupdate'!
It's worth asking for details from the Plesk folks themselves, IMHO.
Not meaning to brag ... just share .. have several school sties that have multiple instances of Moodle on a single server.(hs, jh, elem, pd, other as example). Takes literally minutes to update/upgrade one of those instances .... about 30-50 minutes for all sites ... and most of that time spent watching the backups before/after upgrades or updating and dependent upon if a backup of moodledata is made at the same time.
Using git, one could cron their own 'autoupdate' (with a sensible time schedule) and be in control of when that autoupdate took place.
2 cents worth... and from one who has had not experience with Plesk or cPanel ...
'spirit of sharing', Ken