The easiest way to track vessels at sea is using the AIS system[1], which is based on VHF transponders that broadcast position, course, speed, and other useful information. This data is overlaid on the chart by navigation software which also provides configurable alarms. In fact, thanks to AIS you can even track ships online on sites like MarineTraffic[2].

AIS transponders are required on merchant ships above a certain length. Like with all technology, this can make people complacent and rely on it too much. As others mentioned, in this case the destroyer probably switched their AIS transponder off (but should still be able to receive).

Even then, the navy ship would still appear on the radar screen (not to mention its navigation lights), but radar is much more noisy since it can be affected by the environment (e.g. rainy squalls or waves).

The above is based on my experience as a sailor.

[1] https://en.wikipedia.org/wiki/Automatic_identification_syste...

[2] https://www.marinetraffic.com/

There are only one or two people awake on lookout in a civilian ship.

Air-conditioned bridge, slowly rolling waves, practically nothing to see or do... If they're on a typical watch pattern, 0230 means they got out of bed an hour ago and had just settled into the watch routine.

Pretty easy to drift off.

...for 3-12 minutes. Then BNWAS flashes the light, then it sounds an alarm, then it sounds a different alarm both on the bridge and in the captain's cabin, then it sounds an alarm that the entire crew can hear...

However, yes, even if BNWAS were operational the watch could doze off for long enough to miss the fact that they were being passed, but not by enough, by a smaller faster vessel with no lights and AIS turned off.