Yes, quartz watches are simpler and more accurate yadayada, but there is an interesting engineering challenge in trying to improve accuracy (1 second/day = 10 ppm!) through purely mechanical means.
Constraints foster creativity.
At the end of the day, he would check if the watch had gained or lost time during the day (compared to the broadcast bell chimes radio stations used to broadcast on the hour).
Based on the observation he would set the watch on its right side or left for the night, causing the spring in the self winding mechanism to be slightly more or less tensioned, leading to a slight gain or loss of time by morning to tune the watch to the precise broadcast signal.
No daylight savings where you live? :)
The most accurate quartz movements drift by around 5 secs a year. Chip scale atomic clocks are accurate to around 37ms a year.
Current chip scale atomic clocks are still a little too big for watches, but some applied creativity might change that.
Personally I don't care about clockwork, but I think having an atomic clock on your wrist would be quite cool.
RFID-style sync to some device with accurate time (probably a phone) would be the best way to keep an electronic watch mechanism accurate on a day-by-day basis.
See:
* https://www.citizenwatch-global.com/the-citizen/lineup/5sec/...
* https://www.hodinkee.com/articles/the-citizen-eco-drive-the-...
They've gone to ±1 second per year:
* https://www.hodinkee.com/articles/citizen-caliber-0100-eco-d...
* https://www.citizenwatch-global.com/caliber0100/index.html
It's said that George Daniels could make mechanical watches that were more accurate than the quartz watches of his day[1], 0.4 seconds per year.
Of course that doesn't really matter if the thing is a priceless work of art that no one can afford to wear day to day but it's still really interesting to know how he did it.
Quartz watches' real advantage is just their accuracy but that they are accurate and cheap to make. There's something rather wonderful about a practically disposable Casio like a Bic Biro.
1. https://www.sothebys.com/en/articles/george-daniels-the-worl...
2. https://www.danielslondon.com/daniels-watches/the-space-trav...