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Worth noting, even once mechanical clocks capable of accurately tracking time by the minute became commonplace, people still didn't really care too much about such precision. It wasn't until the 1800s, with the rise of the railroad, that people really started to care about precise timetables and standardized timekeeping across different locations. Before then, every town and village would've had its own slightly different standard for setting the clocks, but this made train schedules a royal pain in the ass, so standardized time zones started to be implemented.
And really, even to this day, many parts of the world (even the developed world) don't really put too much emphasis on precise timekeeping. Punctuality in a lot of places is more a matter of "somewhere in the general vicinity of X time", rather than "X time on the dot". This is a lot closer to how people approached the matter of timekeeping and schedules before the era of rail travel.