@Steve Freides - That is an interesting way to place your non-working hand (or lesser working since we all know that law of irradiation!). I may have to try this once or twice to see if it works. I personally place my hand flush with my body but I can clearly see the benefit of being able to really tighten up the other arm for more strength. Great stuff!!! Have you considered trying resistance bands to assist your full OAPU for a short time? I found that doing this for chin-ups...etc really helps with allowing more volume for practice and when I finally take it off they come far easier.
@Kozushi - Your perception of a military press vs a handstand press are very misguided. Indeed the standard barbell military press involves the leg mucles, but so does the handstand press! A few other implications that are not considered is the range of motion. On a standard wall-headstand press your ROM is from the top of your head and up. Try changing this to a full ROM and then decide if the barbell is easier or harder. Second, you state that a wall headstand is not using the legs because you are leaning against a wall. Consider if you equate the standing military press to the handstand press they both must go through the same range of motion AND have the same amount of support. So you would be equating a full ROM, unsupported handstand press to a basic barbell military press?
I am personally a former Strongman competitor turned calisthenics practitioner. Currently I am 5'6" and 240lbs and I have frequently put more than my own bodyweight above my head with logs, axles, barbells...etc and I can say for certain the handstand press if by far more difficult (even with just standard ROM). I believe the main difference here is because you're able to brace your own body against an external load versus having to balance your body while moving it over a distance. Visually the mechanics are similar but the true specifics vary greatly.
Lastly, I have read from the book "Building the Gymnastic Body" of how gymnasts were able to pull 2-3 times their bodyweight in a deadlift without ever touching a weight previously. I have also watched those YouTube videos where they pit different fitness styles against each other. The one I mention has calisthenics only vs a power builder. While the test was a little one-sided, the calisthenics guy only lost by 1-2 reps on the last exercise. I personally find that while barbells may provide the quickest route to muscle building and strength, it may not be the best. Calisthenics takes more time to develop strength but you have the added benefit of allowing your connective tissue to keep pace. If I could turn back time (NOT a Cher reference), I would 100% devote all of my time developing calisthenics strength before even dreaming of barbells.