I think what this really suggests is that the transformer and power supply are providing multiple voltages. The +9V rail is broken, but the other rail (maybe +5V) for the LED display is fine.
So the alarm chip and the alarm sounder is running only from the 9V battery. If the battery is weak, then the sudden power demand from sounding the alarm is too much, the voltage dips and the chip resets.
There's a reference design in the LM8560N IC PDF that suggests that the additional secondary winding is only used to drive the display, and all other electronics is powered off the main rail.
The 10V cap is kaput and it can't filter the input power, 9V just acts as a filter (even a discharged one). The OP should replace the cap with a 16V one, because a 10V cap on a 9V rail just isn't right..
I was born 1990±2, but I have always been an old soul. I've been waking up to BBC Radio 4 since I was a teenager.
My current one used to belong to my dad, it's a Sony dream machine with FM/AM, circa 2002.
Another alarm clock had been built by my father, some 50 years ago, from medium-scale integration TTL circuits, before the apparition of dedicated integrated circuits for clocks. It still worked perfectly, and because it used a thermo-compensated quartz oscillator it had a much higher accuracy than the cheap quartz clocks or watches that can be bought now.
Since then, whenever I need an alarm or to know the time, I just use a smartphone or computer. I was born much earlier, so I have used dedicated alarm clocks for several decades, but I do not like to use superfluous things, so I no longer use dedicated alarm clocks (or wrist watches or TV sets etc.).
That does not mean that I do not like clocks, either mechanical or electronic. I like them, but I do no longer need them, so I like to examine them like I like to examine some ancient sword, which is beautiful, but for which I do not have any use in my daily life.
I don’t allow anything more complicated than that, a light and a book in the bedroom.
Anyhow, I had to update how it powered the internal speaker (which I was taping into) because it was a little bit too much and the thing would reset.
I’d go into more details but it was trashed and it’s been about 20 years since …
But some other commenters seem to have a good theory that it's a failure in a component
I have also lived in a country where blackouts quite rare, usually just local blackouts due to a tree hitting a powerline, so in any given year I wouldn't expect one.