Since it's no longer that easy to force YouTube to switch to the Flash player, I've used an old version of Firefox (22.0). YouTube defaulted to the Flash player, the same old player before the recent update.
I've switched to the HTML5 player using youtube.com/html5 and YouTube displayed the new player, just like it does in the latest versions of most desktop browsers.
It's obvious that the Flash player has been deprecated and it's still used for older browsers and a small percentage of premium videos. If you're using the latest version of Chrome, Firefox, Safari, the HTML5 player is used by default and you can't even switch to the Flash player.
For some reason, the system requirements page from YouTube's help center still mentions that you need Adobe Flash Player to watch YouTube videos.
