1) The troll made a sweeping generalization about analog vs digital, which is not the same as mechanical vs quartz. Analog watches can be quartz or mechanical. Breitling makes chronomtre certified watches based on quartz movement that will be more accurate over time than an everyday quartz based digital watch (barring radio update, the time keeping equivalent to GPS-updated INS guidance).
2) As Jarhead pointed out, chronometre is a specific category of accurate timepieces, which is why I was so specific with my example to counter the troll.
3) Omega and Rolex make analog chronometre certified watches that are self-winding mechanical. Accurate mechanical analog watches are out there, it's just that the trend is $$$ required, hence the rise of radio compensated watches with inexpensive quartz innards.
4) I don't know what other people do, but I use my watch for rough TLAR timing and as a backup... and the jet for the real thing.
I think that about covers it.