The logic circuits might have been 5V, but I don't think it's save to assume the alarm circuit was. I think it's likely the display driver required more than 5V, so there was higher voltage available on the board. A quick search for component piezo buzzers show they're commonly DC powered and don't appear to need a driver circuit. If there's higher voltage available and you want it to be loud (remember this is an alarm clock), I think you'd want to tap into whatever the display driver is using, not the logic circuits.
I would have to see the clock to tell for sure. I assume it would have a regular piezo element driven by an oscillator on the clock board.