I don't understand - all actual purchases go through mandatory, non-customizable Apple UI. There's a "Confirm your In-App Purchase" AlertView, which will also trigger the entry of your iTunes password if you haven't purchased recently. It would be very difficult to misunderstand the purpose of these dialogs. Am I missing something?

You're missing the fact that if you install any app, even a free one, you will be prompted to enter your password. Then for the next 15 minutes you wlll not be prompted again as your child takes the phone to play their new game and racks up thousands of dollars of in-app purchases.

This was a big story not that long ago and prompted an FTC investigation. Apple fixed it in their latest iOS update by prompting for the password every time.

Even for an adult, using the phone alone without a child's fingers prodding randomly at the screen, a popup with an instantly charge my credit card button and a cancel button right next to each other is hardly a peaceful and serene thing to deal with when it pops up unexpectedly as you try to hit the tiny button that you hoped would allow you to avoid this very question. That's why it reminds me of those "You are trying to leave this webpage! OK Cancel" things.

They have fixed this in the most recent iOS update. In-app purchases will always ask for passwords now.

Yes, that's what I said in the second sentence of my second paragraph above. But since you bring it up again I'll point out that this update is not available for my iPhone 3G that I have retired and now use exclusively for kids games. Instead I have to manually sign out after an install.

Concerning in-app purchases - I hit the same experience with Angry Birds. The text for their in app purchases reads: "Stuck on a tricky level? Out of levels to Play? Bring on the Mighty Eagle!" Not knowing what this was I clicked on the checkmark. Because Eagles are awesome, who wouldn't click on that! Whoops...