Embedded YouTube videos don't play on iPad (iOS 7) while HTML5 search input is visible
Just experienced this first-hand myself and wanted to add my kudos for you having written this up. Your SO question, even without an answer, pointed me in the right direction.
In my case it was nothing to do with Youtube. I had a page generated by Drupal in a large-ish site, in which the site-wide search mechanism used an auto-complete drupal module which had type="search" as the main input's type.
In IOS, users reported that Facebook, Twitter and Google Plus's respective "like" buttons all didn't work, along with a much larger angular app embedded in the page. They all used iframes and none of them seemed to respond to clicks.
Changing this seemingly innocuous, unrelated input's type from search to text solved this problem immediately.
Baffling.