That is the exact reason why I wouldn't use the auto-update feature. I just thought about setting it up to check for updates and give me some sort of notification. I just feel like a reminder every now and then helps me to keep everything up to date and avoid some sort of never change a running system mentality.
Your idea about setting it up and only letting it run occasionally is definitely one to consider. It would at least avoid manually checking the releases of each container similar to the RSS suggestion of /u/InnerScientist
No, none of my containers are exposed to the internet and I don't intend to do so. I leave that to people with more experience. I have however setup the Wireguard VPN feature of my router to access my home network from outside which I need occasionally. But as far as I read, that is considered one of the savest options IF you have to make it available. No outside access is of course always preferred.