NavList:

If you know the time and can see a star at the zenith, then you can indeed determine both your latitude and your longitude from this single event.

But there are practical difficulties.  The zenith is not marked on the sky; you really only have the sea horizon as a reference (or spirit leveling if on land).  If the star is the same angular distance to each part of the horizon, then it is indeed at the zenith, but this takes time to verify, and meanwhile the star is moving.  Also, it is unlikely that any bright star will be within an arcminute or so of the zenith, so the available navigational events are severely restricted unless you can deal with very dim stars.

So it's better to work with several of the things you can observe easily, namely, angular distances of various bright stars to the horizon, observed separately over the course of a few minutes.  A star not at the zenith gives only a "circle of position", not a position directly (as you don't know the star's azimuth to any accuracy), so you need several of these observations.

But, as you say, computation and instrumentation are moving targets, and it's always interesting to see if they can map onto the classical navigation problems in new ways.  Sketch out a complete version of your zenith scheme and try it under the real sky.  (Using a camera, you can see dim stars very near the zenith.)  This kind of playing around can provide, I think, some good intuition about how the classical techniques have been shaped---maybe better than reading a textbook treatment of the techniques already fully formed.

Cheers,

Peter