Okay, my suspicion was correct. This is good news; it's really easy to address — it's even written in the documentation.
Take a look at the screenshot of the Manage Backups page you sent me. The Status of the backup archive is a cloud icon on teal background. If you hover over it you will see it says Remote, i.e. the file is only stored in remote storage (Amazon S3, according to your log file) and no longer present on your site.
As explained in the documentation (click the Help icon you see at the top right of your screenshot) you cannot restore a backup archive directly from remote storage; it needs to be present on your web server.
Click on the Manage remotely stored files button for the backup you want to restore. It is at the right hand side of the screenshot. There's a fetch back to server button. Click on it and wait until it's complete. This will take a while since it's a nearly 3GB backup.
When its done, the status of the backup will change to a checkmark on green background. If you hover over it it will tell you the status is Complete. This means the backup archive is stored on your web server.
Now you will be able to select the backup record and click on Restore to restore it.
After restoring your site please remember to delete this backup archive from your server. It won't happen automatically.
Nicholas K. Dionysopoulos
Lead Developer and Director
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