It definitely wasn't just Apple. Google and Mozilla certainly played a huge role, and I definitely consider them much better web citizens than Apple (Mozilla best of all, obviously), but a lot of sites switched to HTML5 video only because they wanted to be watchable on iOS devices.
Adobe could have struck a deal there, given Apple their 30% and only allowed blessed scripts through the App Store. But they didn't. I would be surprised if there weren't neogiations about this, but in the end Adobe refused.
(Other scripting languages budged, which is why you can find Unity games in the App Store, but not Adobe Flash games. Unity was also banned at the same time Flash was.)
So if anyone kept Adobe Flash off the iPhone, that honor should probably go to Adobe itself.
Google used the strong weight of Youtube to make sure all web browsers implemented a replacement, a long and tiresome process that took the better part of ten years. They bought companies and gave patents away along the way.
I'm not sure exactly what Apple did. Please enlighten me.