Softaculous

The backup filename format could be much better. Rather than e.g. 15.gz, let's see something such as id-hostname-time.gz, e.g. 15-server1-mydomain-com-20160329030000.gz. This will help when identifying backups for disaster recovery.

The user should be able to access backups of their VPS and restore themselves

The backup logs should appear on their own page in the admin panel, not under settings (seems a weird place to put the logs). Better yet, have a logging page in the admin panel which allows the admin to access various logs including backup ones.
"This VPS does not exist. Unable to restore to a non-existant VPS" - lets have a solution to this, rather than having to re-create a VPS with the same specs manually. A GUI which reads the .inf file from the backup and automatically creates a new guest with the same spec would be helpful (let the admin confirm and tweak settings first). In short, a way to restore a backup after the guest has been removed (seems like this would be a common use-case).
The user should see feedback when a restoration is taking place, during restoration the user should not be able to take any action on a VPS (instead a message e.g. "Restoration in progress... 55% complete). Perhaps a simple solution to this is for the VPS to be placed into "managed by admin" mode temporarily (at least this is what I would do for now when restoring, to stop a user from breaking anything).

The backup folder name is the date on which the backup was created isn't  .
We use the backup file name to detect the VPS on which the backup is to be restored.

Understood, however you could still place useful information (such as the VPS hostname) in the backup filename. The ID can still easily be parsed from the filename. e.g. in date-hostname-id format, you would split it up and take the last value (any dashes in the hostname can be removed, it's just to help with identification for the system admin).

To help explain our point of view, we sync these backups off to cloud storage by the filename for disaster recovery. So it would be helpful if they were named in a way that not just Virtualizor understands but humans too.

Thanks for taking the time to reply.