Youtube background play being a paid feature is so brazen.
Google are disabling a basic UX feature in their OS and then letting you pay for it in one app that they control.
The same is true on iOS, but using the media center instead of the media notification, in which case it will work.
On Android native apps this isn't really true at all, but you should use a background service to handle playback to avoid getting killed by the OS on low memory situations. This does not work for YouTube on the web, because they simply use the web visibility API to stop you from resuming the video if you try, unless you have premium.