I tried this a few months ago. Many features were missing then. I couldn't set up backups, and connect directly to `psql`. It was such a pain. I don't know if they've addressed those issues yet. It'd be a hard sell for me to store a database system for a production app in a Docker image.
It's better, but still beta. We handle backups and you can connect directly to psql. We also added migration support as part of app releases. The global distribution feature, though, works with any database, not just our postgres.
Between you and me, we tried really hard not to build a Postgres service. We're committed now, it's gonna be good. For future DBs we plan to get big enough that people like Timescale to ship their products on Fly infrastructure.
This might be in the docs somewhere, but can you bring your own postgres image to your managed service?
You could run your own PG by modifying that app. Right now we're calling it "automated" and not managed, though. All alerts about health and other issues go straight to customers, we don't have DBAs that will touch these things yet.