Ah - that's got nothing to do with supported features. Apple has always been a major backer of web based video distribution - a lot of the tech (from video formats to delivery platforms like HTTP Live Streaming to the tag were partially or even fully invented by Apple.
Your video wasn't working because the by default Safari assumes (correctly) that most video on the web is an ad. Safari generally only tolerates text/image ads* and to get video to work, you need to make it clear to Safari that the video is a real video the user wants to see.
Safari also silently blocks something like 99% of cookies... only cookies that behave like login/session/etc cookies are allowed. That's a lot more problematic than blocking video... since there's often just no way around it.
(* even text/image ads are barely tolerated... as far as I know, Safari is the only major browser that includes explicit support for ad blockers - Chrome/FireFox/etc allow extensions to arbitrarily manipulate the page, but safari actually has an ad blocking API - though they call it "content blocking").