To me this isn't about much else besides safety. For lack of a better word, trans or non-trans we don't have much problem with people trying to "punch up" in sports as long as they can hang. Example - there are women competing in men's sports (HS girl in wrestling or football for example), but even that is rare. To my understanding we already don't allow it going the other way - HS boy doesn't get to join the girl's LaCrosse or soccer team.
To my knowledge that is us acknowledging that our bodies are different and there are safety issues. (Again, I am speaking in generalities and I realize there are exceptions). Muscle mass is different, bone density is different, body structure, hormones, on and on. Males are stronger, faster, hit harder, etc.. and there are injury concerns there. Somebody brought up competitive balance and I think that is secondary, but it's there. Even in a sport like running - isn't the women's world record for the mile 4mins and 15 secs or something? If I remember a recent podcast correctly they brought up something like it's on the level of 1000s of males have gone below that. So I get the point of fair competitive balance too.
Anyway, not sure where that leaves us or how to go about it. Mostly my way of saying that I think it's OK to say that men transitioning to women might not be allowed in some of these competitions. Not because they are trans, but because it's how we are already doing it because of safety. That said, I am open to hearing arguments about timing of transitioning or how hormone therapy in a younger person would quell some of these concerns going forward too. Example - there is probably a bit of a difference between a boy transitioning in his teens and competing later on in life (college, whatever) vs. Bruce Jenner training at a high level through adulthood and then transitioning and competing with women a year later.