The recipe is pretty much 1 of the 2 I use. The other one uses a bit of sugar, which I always use anyway when proofing the yeast, and some yogurt. The yogurt makes a softer, fluffier pita.
Don't add all the remaining flour at once. As noted, you want a soft, shaggy dough. You can always add flour, but not take away. Keep the dough soft but workable.
I bake at 450 F on an upside down heavy sheet pan. I turn the oven on a bit and let it heat for 10-15 minutes after it comes up to temp.
Also, I didn't see this when I looked through.
Slide your fingers under the pita and gently pick it up, then flip it top side down onto the hot stone/baking sheet, and bake for about 3 minutes. The pita will start to puff up after a minute or two and is done when it has fully ballooned. Cover baked pitas with a clean dishtowel while cooking remaining pitas. You need to cover them with the towel so that they will steam and remain soft. They may become brittle otherwise.
I don't know why placing them on the baking sheet as above matters, but it does. You don't really need to flip because it should be already ballooned and only a very small portion will be on the sheet pan. If mine don't poof then I will flip just to get some browning. If they don't poof within the first few minutes they aren't going to later.
My success rate is generally about 80%-85%. And like
@Aunt Bea wrote, experience and repetition matter. I've noticed if I don't make them for a while then they don't come put as good.
I did roll them a little smaller last time so they were a little thicker but they still poofed fine.