It is also worth noting that most alarm clock chips not only keep time by counting AC cycles (often with fallback to 9V battery and suitably divided 32768Hz cry

It is also worth noting that most alarm clock chips not only keep time by counting AC cycles (often with fallback to 9V battery and suitably divided 32768Hz crystal) but also derive timing of all other internal functions from it. For example when LED display is multiplexed into two sections (as it often is) there are no common electrode drivers and instead the display is directly connected to two 1-way rectified taps of mains transformer. Economies of scale in AC-powered radio clocks are such that center taped (or even multi-taped as required for things like LM8562, which seems common in European clocks) transformer is significantly cheaper than two additional transistors. And LM8562 and this timing of everything off mains is the reason why many modern alarm/radio clocks have that convoluted interface for setting time and alarm with two buttons that decrement setting by one minute and increment it really fast (ie. 50/60 minutes per second). [Edit: 50/60 minutes per second, not 25/30, datasheet says that this rate is not dependent on rate of pulses on the setting pin, but I think that nobody really tested what will happen if the button is connected to anything other than mains frequency]