I think it's important to note that not all transgender people consider themselves to have changed their biological -sex-. I think this is exactly the key point J.K. Rowling was trying to make. It's fine to change one's gender identity but one's biological sex is set at birth and at the present time, there is no way to change that.
I'm sure you know that there has been some integration in recent years in regards to combining men and women facilities, as well as creating, say, restrooms where only one person can go at a time and that person can be either gender. Other than 'one person at a time only' options, both options have risks. On the one side, we have people fearing for the safety of women if transgender women are allowed to use female restrooms and on the other hand we have fearing for the safety of transgender women if they are required to use men's restrooms.
When it comes to prisons, it's also complicated. The following news article is a good example:
N.J. trans prisoner who impregnated 2 inmates transferred to men's facility | nbcnews.com
Trans ignore the matter of biology as it cannot be argued and instead try to say that feeling like a woman is the only criteria.
I believe you're right that -some- do. Clearly, not all of them do. There's a documentary you may have heard of or seen called "What is a Woman?" presented by Matt Walsh, where he interviews a transgender woman who makes it clear she does -not- consider herself to be the same as a biological woman. I certainly don't know how many transgender men and women feel that there are some clear differences, but regardless, I think you, me and J.K. Rowling agree that there are.
That simply doesn't fly with actual women or girls or guys for that matter.
Here is where I have a bit of an issue. I'm ok with transgender women being called women, but with the understanding that they are a type of women that can't reproduce, for instance, and also have some other differences, which can generally make them better at sports, for instance.
For those pretending to be the other sex it does present a problem for restrooms/locker rooms.
I think describing trans people as "pretending to be the other sex" isn't quite right. While it's true that I only currently know of one transgender person (the transgender woman in Matt Walsh's documentary) that made it crystal clear that she did not consider herself to be the same as biological women, I strongly suspect that she's not the only transgender person who feels this way.
But as its the one putting themselves in this situation, its their risk to bear. Same as those who prefer motorcycles to cars must bear the risk of far greater harm if they find themselves in an accident.
Speaking as someone who used to ride a scooter, I can certainly attest to the fact that they're more dangerous. Luckily, I never experienced any harm myself, but there was one time that was close. I rode a scooter because it was bought for me as a present from my family. I would have preferred a car, but cars were essentially outside the budget, so that was that. My father had no car at the time and hasn't owned a car since either.
I think the issue of where transgender people go to the washroom is simllar. With enough money, I'm sure they could make it so that they have their own washrooms or what not, but I think most don't have this option.
School kids are a real issue but my wife's high school came up with a great solution. She ran the clinic which had it's own restroom and trans kids (and other kids with medical reasons for needing extra privacy) would just go there. Worked out very well. The trans kids were in there a lot anyway if they were having rough days so they were comfortable with the missus.
That works

. I think it's things like this that will bridge the gap with these types of things.
The real issue is mental health professionals pursuing the vastly more profitable route of unsafe drugs and elective surgery over curing the mental illness. The growing ranks of young people coming to regret decisions made before they were prepared to make such life altering decisions really should be all legislators and medical people need to not allow these treatments before age 21.
The issue of mental illness is a rather touchy one. To this day, some people believe that homosexuality is a "mental illness". It's not something I agree with. In regards to drugs and surgery, I certainly agree that these things should probably not be allowed for minors, but I think that 18 might be old enough.