NavList:

Regarding the tables included in HO214 for interpolating for latitude and hour angle, HO249 effectively includes the same tables in disguise.  "Altitude Correction for Change in Position of Body" interpolates for a difference in hour angle (just convert angle to time), and "Altitude Correction for Change in Position of Observer" can be used to interpolate for latitude by pretending to have moved on a course of 000° or 180° (find a combination of speed and time that gives the desired "course made good").  HO249 even includes alternate versions of these tables at 0.1' precision for reasons I never understood...

As for the advantage of using such tables to interpolate to a particular DR position, I'm not sure there is one. 

For one thing, the point of these "single-entry" tables is to save time at the expense of space on a bookshelf.  The more time and "processor cycles" you spend using these numerical methods of interpolation (as opposed to interpolating graphically on your chart), the less advantage there is to using these tables instead of a far slimmer, multiple-entry table like Davies', Ageton's or simple trigonometry tables.

For another, it is based on an assumption that declination, hour angle, and latitude can all be interpolated for independently of each other.  The whole process treats the line of position as a straight line on a flat surface (i.e. plane trigonometry) rather than as an arc of a small circle on a spherical surface.  The results you get will be "good enough, probably" in most cases, but drawing on a proper chart can produce results of the same or better quality in less time.

The only common use case I can guess at is "The Navy trained me on these tables and I've been told to keep the chart neat."