Yes, we used to use plug-ins for video. This worked reasonably well for quite a long time. I do think calling those plug-ins non-standard is a bit of a joke, given that Flash had 90+% market penetration for years and ran acceptably well almost everywhere, putting it far ahead of any browser in terms of portability and compatibility. By any reasonable definition, it was the de facto standard for on-line video streaming.
Then Apple decided not to follow that standard and broke the Web for their users. For those users, a lot of major sites still "don't work properly" today because of Apple's arrogance. Browser makers are playing along anyway, so now we have an HTML5 video ecosystem where the closest thing to a universally supported video format is patent-encumbered anyway. It is still supported on fewer systems than video using Flash players used to be. And instead of security flaws with your "non-standard" plug-ins like Flash, you're inevitably going to see security flaws with whatever media extensions you install instead (unless you think low-level, natively executed code written in error-prone programming languages and provided by the same kinds of organisations who used to provide the plug-ins is somehow magically going to be secure this time around, in which case I know a Nigerian prince who has a great deal to offer you).
This isn't progress. In what is becoming a serious and recurring problem with the modern web, issues other than what gives the best experience to users and makes things easiest for content providers to provide content are dominating the discussions. Many of the proposals are advocated by other parties with vested interests. And in the end, the result is that less stuff works, and stuff works less well, than the "legacy", "insecure" standards that have actually done the required job pretty well for a decade or two.