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I would love it. But: postgres, +backups, +replica managment. Basically a managed database for postgres in a container.

Check out https://thin.dev/ :) It's similar, supports self-hosting and uses postgres. Quick demo video here: https://www.youtube.com/watch?v=-jj19fpkd2c&t=3s

(I'm founder of Thin)

You may want to check Supabase or Nhost in that case.

PocketBase was specifically designed to be self-contained and running only on a single server.