A doctor identified your sex at birth and the words man and woman go on to identify that sex throughout your life.
A doctor looked at a few phenotypes and made a designation. That is factually what happened. No one anywhere at anytime has assessed me biologically to determine I am man beyond a set of phenotypic expressions. People use a small set of phenotypic expressions (plus electrolysis) to label me as a woman now. That is an ordinary practice for using the terms 'man' and 'woman'. There is nothing incorrect about it.
The point here is not that transgender people and allies are arguing 'man' and 'woman' is defined strictly by appearance. The point is constantly repeating 'biology' is a nonsense dismissal. If you reject the concept that gender identity is meaningful and that the terms 'man' and 'woman' are meaningfully applied based on gender identity, so be it. But if you're going to keep up the routine that a doctor looking at my genitals as a baby is the reason why transgender women aren't women, you're not really making a relevant (or even true) argument.
When I go to see medical doctors, they address me as a woman because it is utterly fucking irrelevant to what they do. They mark 'F' under sex. I will at some point be screened for prostate cancer simply because I have one. I will be screened for breast cancer because I fall within female category of elevated risk. When I went to the ER after surgery, they went for the gynaecological cart because it had the appropriate equipment. There are more examples but it gets increasingly personal. All that practical biology without one fucking whit of care to whether I am designated a 'man' or a 'woman' because those terms do not matter. This wholesale 'man' or 'woman' is not relevant. They do what is specific to my biology which, if we generalize, includes treatment and care which is applicable to men, to women, to both, and in very limited cases what is specific to MtF transexual women.
The terms 'man' and 'woman' predate the biology you are using to define them. You have the relationship backwards. We used biology to describe the terms we had, not the other way around. Those biological descriptions do not perfectly align with how we label people. That is reality. You are talking anachronistic bullshit.
And when it comes to advantage in competition, it is variable to the individual. Not all transgender women have biological advantage. It depends on a number of variables. Some transgender women do have a biological advantage. Some cisgender women have a comparable advantage. If you want to regulate advantage, regulate advantage. NCAA, elite-level sports and professional sports, by all means.
High school is more complicated. I do not know what the correct solution is. But again, not all transgender girls even have an advantage in the first place, so banning transgender girls as a rule makes little sense. Such proposals aren't acknowledging biology--they're being conveniently selective.