At the point you care about backups and standby nodes, you're already sort of outside the cheapo hobby tier.
As for getting the $5/month option: Digitalocean Droplets have a 1-click Dokku install, and you can use https://github.com/dokku/dokku-postgres to instantly pop up a postgres instance.
That's kind of fair. I've been doing some thinking since posting this, and I've been sort of debating what a "reasonable $5/mo version" of this would be. The more I think about what managed database services usually entail, the more I concede that my original request probably isn't really a viable product (unless it was a "free tier"-style loss leader). Like, disk space and bandwidth aren't really the primary cost involved.
I think what I really want is managed backups that live outside of my box, and the ability quickly restore from one if my database crashes or becomes unusable. I think automated failover to another node may not be a reasonable ask for such a cheap product.
Of course, at that point: I could spin up a 1GB DigitalOcean volume for $0.10/mo, and set up scripts to run `pg_dump` every couple hours, clean the volume of older backups to free up space, and ideally a script to reset the database to a given volume. _That's all stuff I don't want to do_, but maybe someone's built a reusable set of scripts for it or something - that Dokku container is promising as a starting point, though I'm a little annoyed it only works with S3(-compatible).
I suppose I could also use DO's Spaces, which is S3-compatible, though also has an annoying minimum (I do not need the 250 gigs you get for $5/mo) https://www.digitalocean.com/products/spaces/
Was about to chime in the same... dokku is great for the hobbiest/cheapish tiers... I wouldn't go below a $10/mo instance, but you can throw a surprising amount of work at it at the lower end.
As for getting the $5/month option: Digitalocean Droplets have a 1-click Dokku install, and you can use https://github.com/dokku/dokku-postgres to instantly pop up a postgres instance.