Why is SuperDuper not running at scheduled time?

[dnanian]

Actually, the manual does explicitly suggest this:

Usage tip
SuperDuper! does not have to be running for a schedule backup to
take place, but the user who scheduled it does have to be logged in,
with their account in front (if Fast User Switching is active), and the
screen must be unlocked.
If your Macintosh is asleep when the time comes to run the backup,
SuperDuper!’s scheduling won’t wake it up, so the backup won’t run.
The way to handle this is to use the Energy Saver Preference Pane in
System Preferences to schedule a wake (or “start up”) event for a
minute before the backup is scheduled to run.

OSX is a bit flaky about how it decides and when it decides to put things back to sleep, but based on my own research one minute is what you need, and that's why I suggested it.