Moodle currently has it's own media embedding code that checks what browser the user has, what media filters are on etc. and writes tailored code for each combination.

The experience might work a bit better for the de-facto standard of H.264 if we used something like video.js to get a consistent experience across the most common browsers and formats and only then fell back to the more esoteric options.

(There's also http://mediaelementjs.com/ but that has a JQuery dependancy.)