You don't need a precise time source to make power grids possible. You don't need any time source at all, as generator operators simply synchronise with the grid's current frequency (and phase) before throwing the switch that electrically connects the generator to the grid. And once the generators are connected, they are automatically locked to the exact same frequency and phase. It's not possible for them to fall out of phase without the electrical connection being broken (If you try to force a generator out of phase, it will draw more current trying to get back in phase and will eventually blow a fuse)
For engineering reasons, it's useful to keep the frequency within a few percent of a standard, but for most purposes, it doesn't matter if the grid is running at 58Hz, 60Hz or 62Hz and you can achieve way more accuracy with crude mechanical governor. Many simpler backup generators use nothing more than a mechanical governor to maintain their frequency.
The primary reason why power grids used accurate master clocks is actually the secondary reason that this article mentions: Automated time synchronisation.
This predates the days of modern quartz clocks. It was possible to make very precise mechanical clocks (especially for navigation use), but they were impractical and too expensive for every day use. The average clock or pocket watch would gain or lose several minutes per day and required constant manual adjustment. Loud bells ringing each hour would allow a town or small city to keep the same time, but different towns would be out of sync with each other.
There were various competing solutions at this time. Clocks would be synchronized over long distances with time signals transmitted over telegraph or radio. Paris actually had a network of pneumatic tubes that drove synchronised clocks driven with a pulse of air every minute: https://www.amusingplanet.com/2022/02/the-pneumatic-clocks-o...
But the power grid neatly solved this problem.
Not only did it distribute power, but it distributed a synchronised time signal across the entire nation. Your complicated mechanical wall clock could be replaced by a simple electric synchronous motor that drove the clock hands and it would keep perfect sync with every other clock on the same power grid.
All you needed to make this useful was a single master clock that kept the power grid running at exactly 60Hz (well, it actually drifts as load varies, but they deliberately vary it so there are exactly 5184000 pulses per day).
This time keeping service that power companies supplied as a secondary effect of their primary purpose was typically mandated by government regulations, as cheap and accurate synchronized time is a boost to the economy.