By saying that chatting and logging are separate processes, and each one has end to end encryption. Only the clients and the archive can see the text.

And that's what the actual quote says. End to end from phone to archive.

The "TeleMessage Archive Server" in the diagram is not the archive. It's a relay that should not have access to the plaintext, but does. And because TeleMessage owns it, they get access too.

The "Archive Destination" is the actual archive and the only thing that should have decryption keys.

This actually seems pretty trivial to me, without a custom Signal client. You link a secure PC with the secure archival software to your Signal account and it will receive all messages E2E encrypted.