iPod Touch Passcode doesn't work after restore from backup

That is normal behavior. Each reinstall of settings on a clean device requires a reboot. It doesn't matter if the device is 'clean' due an iOS upgrade or clean from a security wipe. iOS backups are not simply copied over. They are interpreted for security and compatibility.

Have a look on the device next time you upgrade iOS. It should perform two reboots. One for the iOS and one after your settings are in place.

Oddly, he should have don't that second reboot on his own since its part of the procedure.

The solution to my problem was rather simple, though I still don't understand it:

Reboot.

My battery drained completely, and when it came back on after plugging it in it presented the expected numeric keypad and accepted my passcode. It seems rebooting was all I needed to get back to normal.

It doesn't make sense to my why it happened that way. If you have an explanation, I'll happily accept your answer.