At a conference one time, Pamela Marie said (paraphrased):
Golden rule asking a question: imagine trying to answer it
Golden rule in answering: imagine getting your answer
I only mention this because I think it's useful for everyone in increasing how good of answers you get you when you ask questions. For your question, it's not immediately obvious what "fail like this" is referring to. I was able to see the problem because you mentioned it behaves as expected in Safari and I could compare and see the difference, so that's good, I'd just error on being even more clear.
I'm afraid I don't know what's going on there. I see the <path> isn't a stroke though, it's a shape, even though it's doing the "draw itself" trick. I think if I was going for this effect I'd make it a stroke on the path and use drawSVG.