What's Old is New Again
I took a Celestial Navigation course at UCBerkeley in the 1960s. We didn't go out to do actual sights, though the theory was discussed, but we did plenty of sight reductions to get a position from data supplied.
As for the time issue...at that point anyone who actually going to do navigation did their level best to afford to buy an Acutron watch, as they were more accurate than the ship's chronometers at the time. So long as it was set correctly and you either knew it's drift and rate or what offset to apply for a limited amount of time, you were set for time keeping good enough to navigate with. Good quality modern watches are better. For that matter, I have a $3 RTC attached to a Raspberry Pi that is good to within about 30 seconds per year and means the time error after a year would be within 7.5 arc seconds of correct, where one arc-minute is one Nautical mile.