Okay, hopefully it will help when I say that I had absolutely no difficulty reading that passage - I never had to go back to clarify what the writer meant the way you describe.
I didn't, either, but it's easy to see how it could be problematic. Take the following sentence:
While Ryan said the team’s reaction was positive, they did encounter obstacles with their decision to use they
That could easily be a garden path sentence, because the "they" after the comma would, in most contexts, refer to "the team". The sentence could easily have gone something like: "While Ryan said the team's reaction was positive, they did encounter obstacles from other teams in the league, who felt they should not be allowed to compete with a non-binary member". That would be an equally valid sentence, and it's only several words after that "they" that it becomes clear who it is referring to and you can mentally go back and put the word in its correct context.
I don't think the singular "they" is the best way to facilitate clear communication. Especially if the aim is to have everybody adopt it. It must be remembered that there are plenty of people out there who find reading difficult (whether due to educational failings or some medical reason), and certainly many who are less literate than professional writers like the author of that article.
I don't think widespread adoption of other pronouns is likely now, for reasons I've already gone in to at length, but I do think that there needs to be some conventions developed for identifying who the word "they" is referring to in any particular context, if that is to be adopted as widely as it is hoped it will be. At least before Millennials and younger have become old enough to dominate culture, society, and institutions. And maybe even Millennials are a little too old for it to be widespread and natural.
I'm not sure what that convention could be, though. A specific sentence construction seems unwieldy and prone to mistakes, modifications in pronunciation wouldn't affect the written word, and spelling wouldn't affect pronunciation. I wonder if perhaps one or other context will mutate over time? I could see "they" mutating into "thy", for example. "Thy", "tham", "thars", perhaps?
Perhaps not. But, as it is, I think there are fundamental flaws with every proposed and adopted solution to the problem of gender-neutral pronouns in the English language. Which is a shame, because widespread adoption of one would be useful - even absent the existence of enbys - and is long overdue, and Sweden has demonstrated how quickly a non-flawed solution can be widely adopted. I fear that without the flaws in the English solutions being worked through that it'll continue to be a long, uphill battle, and would be even if people who object on ideological grounds didn't exist.