They do. In your App yml you can set Discourse to Stable Branch. However you still need to once in a while login to the server and upgrade.
This is the benefit of paying for a hosted plan is that you don’t need to worry about crashes as the Host plan provider looks after it
Self+Hosted has benefits but you may run into trouble and if you are unable to fix it. May need to pay someone to correct the issue(s). This is why making backups is important. Especially with a major update. Your VPs provider also has an option to make a snapshot of your server that you can restore if an update via say command line fails “Spectacularly”.
In my journey now if just over 7 years I have had to learn many new things. Most recently spun up a new server and used Rsync to move my forum to a fully updated current Ubuntu server. Old one was iirc was Ubuntu 20.x. Now running the latest iirc is Ubuntu 24.x. had I not taken the time to learn and complete this U would have eventually had a “Spectacular” failed upgrade.