Having had time to read more lately, including English history, I think this 'trad' male/female thing is a more recent invention, say 200 years or so. The dainty female was a late Victorian thing amongst the middle or upper classes.
Before that say from about 1300 to the industrial revolution a woman would be expected to be able to carry on her husband's occupation, including being a blacksmith or armourer, in case he were called up to fight in a war. Life was a lot harder and worrying about 'roles' not top of the agenda. Traditionally, everybody had to muck in to survive unless you owned the local castle.
This was visibly demonstrated to me last Monday at a confirmation service where I musicianed with a small band (as in brass). 'James' is now 'Kate', and this is the first time I had seen him after learning of this supposed transition a few months ago. He simply doesn't look like a girl, and I doubt cosmetic surgery is ever likely to change this. The candidates were all girls, and the difference was obvious. I think the pretence is cruel. I don't know the background to this, there may be a genuine medical condition underlying it, but this is almost certainly not the way to deal with it, and my conscience will not let me 'affirm' it, particularly in the context of last Monday. It's not loving in the agape sense of the term.
I think you are absolutely right to argue against men identifying as women from being allowed to participate in - and ruin - women's sports. Even if you want to argue about gender and roles, and I have a fairly laid back view of roles, physically nothing really changes. The idea of transitioning imo is a myth, based on a lie. I find it astonishing when feminists, of all people, argue in favour of it when it comea to men supposedly transitioning.