Would it be possible to just run it ourselves now from a docker image like https://hub.docker.com/r/bitnami/pgbouncer/ ?

Even better would be a Postgres plugin store that allows you to install plugins and configure them from the dashboard. This could be a plugin on it.

I think this is a key feature these days, many developers are opting for the serverless stack or JamStack path. Supabase for example offers this in their PostgreSQL service. It would be interesting Railway to introduce this functionality.