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You might be interested in ZeroFS [0], it works great with Postgres [1].

To achieve what you describe, you should be just able to setup a Postgres replica that’s setup on top of ZeroFS.

There is wal-g that moves the wal files to s3 and you can spin up any number of instances off of that. Works great for catching up secondary servers

just a quick look on it, is that more like pg_dump or barman?

Did Postgres's WAL archiving feature not work for you?