Digital clocks are decades old technology; why don't (some) keep proper time. Without syncing? Hm, now I'm interested in how they work ...
So a electronics manufacturer takes this little tiny piece of quartz crystal and put a voltage to it or near it. This distorts the crystal slightly. The crystal is piezoelectric, meaning when you stress it by bending it, it makes an electrical charge. So it pops this charge out, and straightens out. It then repeats. The output of these little pulses is a frequency, based on the crystal, its shape, impurities, etc. they shave it with a laser to make it as precise as possible for whatever frequency they're trying to make. common watches usually use 32,768 Hz.
The clock counts these little pluses like the tick of a clock.. the clock counts 32,768 of them then adds 1 second to the output. super simple, based on the crystal oscillator.
Of course, when you're counting this TINY time between little ticks, the smallest of errors add up over time, and thats why cheap watches and clocks drift off.