Depends what you define as "accurate". How many seconds wrong over, say, one month do you consider accurate enough?
Such clocks virtually never try to use the 60Hz supply as a time reference. It is not good enough. The 60 Hz frequency does alter slightly from time to time, although not a lot and only for short periods. But what happens when power fails? For a period of minutes to many hours there is no 60 Hz signal to use. So, you need some timekeeping mechanism that is accurate for a significant time period before you get back your reference signal. If you can do that well, you don't need the reference signal in the first place.
MANY such systems now are built using quartz crystal oscillator chips. A suitable crystal of quartz (quite small) cut to an exact angle will have a VERY stable known natural oscillation frequency which does not vary much over a reasonable temperature range. So using that, the chip operates as an oscillator with a frequency tied to that stable source. This signal never quits unless the entire circuit loses power, in which case the clock it drives is dead and loses its settings. To guard against even that, many such systems use a small battery to keep the oscillator and its digital circuits live over extended time periods. Those circuits use very little power so a battery is suitable, and the higher-power circuits used to display the results are left to die in a power failure scenario, then re-start and display the current data when the can. These systems are so good they are better than many possible external references.
Even systems now that use external references (for example, to a clock signal from a cell phone service, or retrieving a time reference by internet from a national atomic clock service) do not use them constantly. They use their own internal stable clock circuits, and from time to time make one-tome spot checks for an update to correct their time reading if necessary.
For a good example of how accurate systems with NO external reference can be, look at modern wrist watches with no Wi-Fi capabilities. The one I have now goes wrong by maybe a second or two over a month. I check it using internet access to our nation's atomic clock signals via my desktop system and adjust it and my computer's clock as needed. This watch, given to me by my son (who knows I have a "thing" about correct time) has no replaceable battery - it's solar powered totally, and must have a small rechargeable battery for night time.
By the way, this is an interesting side note. Our national time service, in addition to making data available on the internet, broadcasts their signals on several radio frequencies. The time readings are "dead accurate" as can be, of course. The audio signal sent out is a series of beep tones. Each beep lasts exactly ½ second, spaced one second apart, and at a frequency of exactly 1 KHz. For each broadcast being sent out, the carrier frequency also is VERY accurate. So those signals contain accurate time and frequency references over several time scales.
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