I guess the only way for all participants to have a fair shot in any competitive sport would be to either introduce two extra categories (I don't think lumping all transgender people into the same category would be fair either, M to F would still have different physical advantages than F to M) or to allow transgender people to compete in the category of their choice with some kind of adjustment. That could be an adjustment that puts them at an advantage (e.g. F to M competing in a male category) or penalises them (e.g. M to F competing in a female category).
However, I'm not sure anyone really wants scores being messed with. I mean, no one makes reasonable adjustments at the Olympics, instead we have the Paralympics. I suspect it's because psychologically, the winners want to feel confident they definitely won outright, not because their scores were tampered with. People I guess prefer to compete with people with the same advantages and/or challenges as them.
So... maybe two extra categories makes most sense. Male, female and then two transgender categories. And with some kind of basic entry requirements for all four categories to stop anyone from taking the piss and entering into one of the new transgender categories for a laugh, on the basis that they don't think there are enough serious players in those categories for that particular sport.
Might take a while until we get enough people competing in those two new categories for them to become something worth watching, but in 10 years' time, the uptake could be high enough for all four categories to warrant equal attention and acclaim from the general public.
With children... I'm not sure what the answer is. Thinking back to before puberty hit, I think there were still clear differences between how the boys and girls did in sport. I suspect you'd still have to have the extra transgender categories. It might make it harder for transgender children to have enough sporting competition though because many transgender people (in today's world anyway) will only identify as such when they're on the cusp of adulthood.
The earlier comments about Rosa Parks are ridiculous, because she was supposed to sit in a particular spot because of the colour of her skin, and segregating people based on how they look makes zero sense. With sport, we're not talking about people look, we're talking about the bits that you can't see - their physiology. It's all about hormones and muscles and making sure people competing against people are doing so on an equal basis.
There are some nasty issues when it comes to discriminating against people on the basis of their race or gender, but they are not the same issues and to treat them as such would be to create an injustice.