Sort of? How it is sort of when you said the same thing I did. Just in a different phrasing.
I said sort of because the apps are "sort of" in the backup. During the restore of a backup, the apps are restored along with the data.
There's no reason to restore again. iTunes could be set not to sync automatically when a device is plugged in and that could be blocking the auto sync after the backup. Or maybe the apps aren't in the computer to be synced by the automatic process.
Assuming the apps are in the computer, plug in the iPad and hit sync. That's all that should be needed. Not another restore.
The setting doesn't have any affect during the restore process. Mine is set to not sync automatically and my iPhone still did after the restore was complete. I'd assume it ignores it since syncing the apps back are part of the restore process.
The setting only applies to devices that are already setup and then plugged in to iTunes.
Yes, if the apps aren't in iTunes, obviously iTunes can't sync them back. This is very possible why the OP didn't get the games synced back. Although, if there's purchases on the device that aren't in iTunes, you will be prompted to sync those upon first plug in.
Follow the steps
1. Connect iPad 3
2. Go to Backup and Select the option that allows you to store in your local computer.
3. Now connect iPad Mini and select which apps you want from your backup.
Note: App Data for most apps will sync. Others using iCloud takes time to sync depending on your internet connection.
Hope this helps!
You cannot selectively choose what gets restored when restoring from a backup. If they do what you suggested, all they will be doing is restoring from the iPad 3 backup which they have already done.