I just realized I'm wrong on one thing. Both Safari and Firefox on mobile will play YouTube in the background if you have Red. If you don't, Firefox won't play it either. But I'm sure Apple broke Firefox too.
Edit: I'm of course talking about real Firefox that uses the real Gecko engine. And Chrome on iOS isn't Chrome either.
Okay, but my point is that Opera can do with without YouTube Red, so Safari could, too, if Apple wanted it to.