Hello @baddog,
Yes it does depend on the browser because each of the browser will render the video differently and most of theme disable the background videos. WordPress Media Elements does handles the video elements differently as well. In your example website, since it is a custom html design, the developers may have found a solution to force the video and trick the browser to autoplay the video.
Videos from Youtube and Vimeo will not be played by the default browser video player. It needs to run in the Youtube app or other video player app in the device. Not just Pro, X or WordPress, Revolution Slider and other themes and plugins does not play videos in smaller devices as of this moment. Sooner or later, this shall be done in the future releases.
Thanks for your understanding.