I had one of these clocks in the past. I remember the clock working without a battery at all. The battery is only necessary if you want to unplug the clock and move it somewhere else without having to reprogram the time.
Heavily depends on how exposed the infrastructure is, I'd say. I currently live in Spain and been experiencing way more power outages than in Sweden, where I was born and raised, even though the weather is much more extreme in Sweden. Main difference, as far as I can tell, is that Sweden mostly digs trenches for all sorts of cables and infrastructure, while Spain tends to (still) hang cables in the air and on the outside of walls. I still get my fiber from a cable that comes hanging down from the front-side balcony of my flat here in Barcelona, and that's the official installation from my ISP.
There's also a US-wide time signal on WWV/WWVB - WWVB having a very simple binary-coded decimal time signal at 35 bits per minute.
Alarm clocks could dispense with the battery entirely by using non-volatile storage for the settings and alarms then using the WWVB signal to set the time. Upon resumption of power it could easily configure itself with no user interaction required.
Ironically my old alarm clock from the very early 2000s supported WWVB (with a physical switch for local time zone) but it still used non-volatile RAM for the alarm so upon losing power it would come back, get the correct time within a few minutes, but completely lose all alarms. Infuriating then and now.