I believe the change into “h” sounds is more applied to words that were traditionally initiated in “p” but morphed over time, in turn also making words starting on “p” an oddity.
For a sound to occur in a language, there are two requirements:
The obvious one is that it has to be possible.
But the less-obvious one is that its precursor in the ancestral language must have been possible there. This is what causes strange gaps where it looks like a sound should be possible, and yet it never occurs.
Japanese words starting with "p" still exist, but they tend to be loanwords (pokemon = pocket + monster) or onomatopoeia (pachinko = pachin + ko). Thinking of it now, pokapoka is also a word, but again, an onomatopoeia.
> It was named after the Japanese onomatopoeic word pokkiri (ポッキリ), which is supposed to resemble the sound of the snack being cracked.
The same is probably true for the Japanese word for flag, 旗/hata - it would have been pronounced /pata/ which is suspiciously reminiscent of the onomatopoeia /patapata/, often used for thin pieces of material flapping in the wind.
This claim seems profoundly unlikely.