Back when alarms could turn your phone on

I see this slide at church every Sunday right before mass:

And every time I see it, I ensure my phone is on silent mode. Today morning though, that slide was put up for way too long, and I couldn't stop thinking: what if an alarm goes off! It's still gonna ring even if the phone is switched off... or will it?

Years back when I'd use a Nokia T9 phone, the alarm would go off even when the phone was switched off. I know because I used to switch my phone off every night, and put it back on only after I was back from school the next day (unless I had to look something up using the phone's GPRS connection).

Phones these days don't seem to support alarms that can wake the phone up! The android issue tracker has the feature request marked as unfeasible because it's hardware dependent. iPhones don't seem to support it either.

The feature probably doesn't make much sense anymore: with airplane mode and battery saver, there's almost no need to switch the phone off ever... except for updates ;)