Yeah I remember seeing those "atomic" clocks in store flyers here too. I guess at the time they made sense but now you can get a reasonably accurate clock with any quartz that you manually sync to your phone or computer time.
Over here we still have Daylight Savings so we have to do that twice a year anyway.
You can not without the syncing.
The point is for the syncing (to the standard time) to be fully autonomous.
Dunno how with the MSF, but here we are receiving the DCF77 and that contains a bit telling if the DST is in effect or not, so you don't have to adjust anything.
There are multiple methods usable to distribute the official standard time:
These transmitters (generally LW) have the advantage of the receivers being of very low current consumption: Typical clock receiver consumes around 100uA @ 1.5V supply (for a DCF77) when active, not much more than the average consumption of a wall clock (the stepper consumes about 5mA for ~20ms pulse, each second; the rest becomes few uA); but it could be active just for ~5 minutes each day, so in average way below 1uA (so insignificant even for small LCD digital clock normally consuming about 10uA).
Then there is the NTP via the internet. Consumes significantly more than the LW receivers, need configuration, but don't need the bulky LW ferrite antenna.
And then is the GPS, the most accurate, but works only outdoors (or with an outdoor antenna), even higher consumption than the NTP via WiFi or Ethernet.
So if you are talking about a battery powered clock, there is not much else usable than the LW transmission, in order to maintain the battery life reasonable.
For a frequency standard, years ago the analog TV was a good distribution (at least here - the public service TV stations used the national atomic clock standard as a time piece to generate all the sync signals the PAL system needed), so hook on any of those signals into any small cheap TV receiver and you had and atomic clock grade accuracy at your workbench. Great mainly for instrument calibration, where the phase noise could be averaged out. Or you need some quartz-VCO based PLL to "clean it up" for other uses.
Today you have to buy a dedicated GPS module for that and use an outdoor antenna. But on the other hand you get pretty clean signal (usually the 10MHz instrument standard).
You may still use these time keeping transmitters, but during convenient "work hours" the signal quality uses to be pretty bad (lot of "industrial" and even atmospheric noise for the LW, bad propagation for the SW). Plus there is a lot of noise, so need heavy PLL filtering.