The most notable result I've seen is a transgender (recently transitioned) MMA fighter fracturing the skull of an opponent in an appallingly one sided and brutal fight.
Denying the most basic facts of hormone effects (muscle and bone density effects of testosterone and then the bone density sparing effects of estrogen) that have not even been questioned for decades is now unfortunately a thing.
Which is to say, we probably shouldn't be grouping athletes by gender/sex, but rather imposing a threshold of testosterone, and having a league for people with (historic) low T and a league for people with (historic) high T.
There'd likely be an interesting opportunity in the middle for co-ed athletics leagues, where the people of both physical sexes that have "average" historic T levels (i.e. who fall on the mean of T levels of all humans) could compete fairly with one-another.
Also, as a tangent: an interesting question brought up by that, is that some athletes actually have an inherent T level that would be considered "using steroids" if a person of average T took TRT to achieve that level. Should such athletes be allowed to compete, if others aren't allowed to achieve their level? (IMHO, it's a bit like a sport where tallness is an advantage allowing people with natural gigantism, but banning people who've been given limb-lengthening surgery.)
Gender has been used as a synonym for sex for a long time. Merriam-Webster and The Free Dictionary list that meaning before the more recent meaning.
You're simply wrong to claim the way you use the word is the only way it can be used.
8:10 and 33:00 https://vimeo.com/19707588
(The whole piece is excellent though)