...the "doubling" of the first Half hitch in this way makes ... sense,
since its use prevents the risk of the capsizing of the first Half hitch,
while the tail is tensioned to make the second Half hitch
In a "series of Half hitches", only the last one is a genuine, proper half hitch. All the others are nipping loops, because both their legs are loaded - just as the nipping loop of the common bowline, the Water bowline, the ABoK#160-161, and the Sheepshank. Of course, if one insists to "see" a half hitch in the bowline, he can baptize every nipping loop a "half hitch", he will discover a relation between the bowline and the Sheet bend, and he will keep repeating the mistake of Ashley during the next millenium.
Now, my point is that "two Half hitches" ( i.e., one nipping loop and one half hitch ) are better than a doubled one - and four "Half hitches" ( i.e., three nipping loops and one half hitch ) are better than two doubled ones ( i.e., one double nipping loop and one doubled half hitch, like the one shown in the picture of kd8eeh )
One may ask : Why then we use double nipping loops, and doubled half hitches ? I believe that this has to do with other things - we do not use them because they nip the penetrating segment more forcefully, but because they form a longer "tube', so this penetrating segment is aligned better - and being aligned better, it is nipped by the next nipping loop / half hitch better ( as in the ABoK#1854-1857 you are referring to.)
There may also be some other reasons : In the Gleipnir, for example, it is beneficial to use a double, or even a triple nipping loop, so the "tube" formed by the multiple coils will be sufficiently long to encircle two twisted penetrating segments/tails - two segments/tails that make half a turn around each other, in the form of a double helix. This enhances the friction in between those two segments, not between each one of those two segments and the coils of the nipping loop...
Of course, this is only a theory, that was initiated by the shock of the Gleipnir : the unexpected effectiveness of the nipping loop, without the help of any collar. I hope that somebody will someday test the gripping power of the single, double and triple nipping loops, and settle this issue once for all.