Actually yeah this is something I'd forgotten about, that is indeed a downside to Firefox.

I had this problem with the British Film Institute's streaming service. I emailed them about it, they replied "please try with Chrome on Windows PC" and I replied "no thanks, please send a refund, I will pirate the film instead".

The other case where I ran into videos not playing was Twitter dot com. I tend not to click links into that site any more though.