If your idea of fun requires women to be subjected to involuntary and invasive medical checks just to exclude some people, I don't care much for it.
It's clear that you don't know what you are talking about.
I don't know about Europe, but in most states in the U.S. a physical is required for participation in school sports and has been for decades. The purpose of that physical is not to exclude a class of people, but to ensure that the person is healthy enough for the sport. There is no blood test done, but the doctor will listen to your heart and lungs, check out your general physical condition and, at least on boys, check for hernia. (Grab balls, turn head and cough.)
No one regards this as particularly invasive. It's done by your family doctor. In the process, all the information generally required to determine sex is collected. (Some intersex conditions may not be detected, but that is not the topic.)
I've had kids of both sexes in go through this. It's done every year.
They are deprived only if you think they are somehow entitled to winning, which is not how competitive sports work I think. There is always a large chance that someone else scores higher.
No one has the right to win. But they do have the right to be competitive in their appropriate class. Classes are determined by the things that affect competitiveness. This includes weight classes in some sports, age groups in most, and sex in most.
The classes are not designed to exclude someone from participating. They are designed to include everyone in the class appropriate due to physical properties. A trans woman is not excluded. They are simply classified according to sex, not gender. Why? because sex affects competitiveness while gender does not.
A case can be made that someone who had transitioned to some point should qualify for the female division as hormones or surgery may have sufficiently abrogated the competitive advantage conferred by their sex. Those are, and should be individual decisions based upon meeting a criteria.
In other words, the "invasive" testing you mention is only required when
voluntarily requesting an exception to the classification system.
Quite a few athletes who had every right to believe they were biological women were deprived of their passion and livelihood when invasive medical tests were used in an attempt to exclude transwomen, but I guess they don't count.
Those tests were not designed to include trans-women. They were designed to detect cheating via hormones and other PEDs. These tests are routine in most professional sports. Males are going to fail the criteria for the women's division due to natural hormone levels, so yes, it will detect sex.
But the bigger scandals, I believe, were along the lines of the East German women who were unknowingly given steroids by their government. The tests are to detect cheating. And of course, Lance Armstrong, Barry Bonds, etc. None of which had to do with trying to exclude trans athletes.
