Well, after some more testing I've come to the conclusion that there's definitely still some weirdness going on that I'd love to tell you if it matters, but the end result is simply that Chrome usually works well. Even though Firefox should also have worked. But as long as there's one way to play it properly, I don't really see a reason to complain.
Thank you for your help, here! Because the description said that Firefox and Chrome were both the intended browsers and I always use Firefox, I don't think I would have gotten the idea to try Chrome. And even if I had, changing the refresh rate might (but might also not, it's all very weird) have affected something, and in that case I definitely would never have figured it out on my own. I know it's probably been a lot of effort on just one player, but I really do appreciate it!