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.
Most places in the world that are slightly more exposed to weather than average run the risk of power outages. That'll be anywhere from Norway to Texas to Australia. In other parts of the world blackouts may be enriched by lack of grid capacity or incompetence.
> Most places in the world that are slightly more exposed to weather than average run the risk of power outages
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.
I had the same experience when I lived in Germany back in the 80/90s. All the infrastructure was underground so the only time we had a power outage was when someone was digging and hit the line. I always bring this up now days living in Georgia where a strong wind can knock out power for a few hours.
It is very annoying because a PC motherboard using a coin-cell battery to maintain the clock can often go years without changing the battery (yes I know many of them will eventually leak but it can take decades).
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.