I haven't read the full thread, but is this the first time you and your brother have had an extended conversation about trans issues?
Is your brother a sporty person? I'm quite surprised about the dismissal around sports as it's the one area that most men instinctively understand, even if they don't get the other concerns. Women are more likely to not care about men in women's sports. I guess it's entirely feasible for some men to not grasp the implications.
Anyway, what I wanted to say was that most people change their minds fairly slowly and over a period of time, especially when it comes to difficult and complicated issues. I think many of us here were very supportive of trans ideology and didn't give it much thought, but over time questions and doubts surfaced as we were exposed to certain troubling concepts.
When you are firm in your own position, having had your time to come to your own conclusion, it is very tempting to dump all this information and frustration on another person in a well-meaning attempt to get them to see your point. The other person though will feel swamped by all the questions and points and simply dig down on their position because they feel pressure to pick a side or feel like their own personal values are under attack.
I experienced this when I started to lose my faith in god and realised that the Jehovah's Witnesses were a cult. There was so much about the organisation that I had discovered and I was so angry and frustrated at my mum and others around me who didn't seem concerned or who were very devoted to the belief system. I want to grab them and shake them and shout how can you believe all this? I wanted to sit them down and show them all the disturbing things I had read about the internal workings of the religious group.
It's a very strong urge and a very understandable reaction. And I get similar feelings with the trans stuff.
However, the best method is actually a sort of drip feeding over a long period. Some people get things straight away. Others take longer. It requires a bit of skill in slowly encouraging a person to reflect on difficult questions and come to a conclusion by themselves. You ask relatively innocuous questions or make certain points on different aspects. Rather than unloading it all at once, you make small points here and there, on different occasions and on different topics. Not confrontational or labelling someone as stupid, sexist etc, just pointing out some absurdities or some things being weirdly unfair.
Some people might never change their mind. But I do think the drip feeding method helps a lot of people to feel able to give themselves permission to start asking questions and over time their position might start to shift. It takes time and a lot of patience though.
Most JWs are afraid to ask difficult or controversial questions. They are afraid to challenge the religious authority because it's viewed as heresy. I think most people who swallow trans ideology feel the same. They haven't given it much thought but they know anyone who questions or challenges it is vilified as a transphobe. So they don't see the point in engaging in such debates or conversations. The instinctive response is to deflect or shut down or avoid anything to do with critical analysis in this area.
The key is giving people room to reflect and change their mind. Their initial response is to be dismissive. Lots of people who used to believe in god were initially dismissive of arguments put forward by atheists, maybe even mocking them or getting angry with them. But over time, as they listened to the questions and points ... And maybe they even tried to counter such facts with their own ... The doubts start to grow... The unease... And they start to question themselves and wonder why they are finding it more and more difficult to justify such beliefs. Until they have to admit to themselves that they too are now an atheist.