Fun story. The actual reality, as it happened: the older woman became a meme, a topic of gossip, possibly moralizing stories, eventually even brought up by commenters of tech-related social media boards.
Meanwhile, as people got used to the clock, the society became used to staying in synch at hour and sub-hour intervals. As better clocks were invented, synchronization became counted in minutes. This simplified things, allowing people to do more in the day and coordinate less, as they learned to rely on agreements to independently do things at specific time. Shops started posting opening hours, school started posting lesson plans, professionals started billing by minutes.
Clocks became widespread, and synchronized remotely - first from the church clock, then over the wire, over the radio, finally over the Internet. The loss of shared time reference at minute-granularity became rarer and rarer, until it became unheard of (except in excuses of those who failed to keep their commitments).
Losing a clock now would indeed throw society into disarray. If all clocks magically stopped tomorrow, many would starve and die before society figured out some way to restore time-keeping by proxy.
Clocks are a perfect example of the ratched effect of technology. A new technology at first helps you, or gives you advantage. As it spreads out more, it becomes relied on. Eventually, reaching ubiquity, it becomes a dependency, and everyone is assumed to be using it.
In a modern society, you cannot live without a properly synced clock anymore.