I want a single place to define where notifications of “Things that need my attention” should be sent to.
These notifications should include: All devices with low battery, all devices which became unavailable, all integrations which became unavailable etc.
First you need these notification destinations and there is no requirement from HA that you should have one. Except the notification in browser that everyone has.
Everything else can be solved by automationss. I guess it is too hard to meet all use cases. For instance in one case 20% battery needs to be notified but another device can run months with 10%. Unavailability is also relative and depends on how often the device should update its status. My powermeter sends me a notification if it has been quiet more than two minutes but my thermometer updates every 10 minute.
WTH: Why don't low batteries show up in the repairs section? has some examples but you may want to vote for that too.
I like this idea, but to help out until it gets implemented, I use this for devices that have dropped off: