No you have not. You have not shown any studies that used control groups.
How exactly do you carry out test with control groups for widely-exercised medical care featuring tens out thousands of people and at a societal level?
The control group is literally the same group of people BEFORE INTERVENTION. If you have a group of people who have a remarkably high rate of, say, suicide or self harm, then you carry out a change in the standard of care of those people and it's discovered that the rate of suicide in that group significantly changed, you are comparing two groups. "Control groups" are generally used for small scale experiments, like drug trials.
You could just as well be saying that if GD kids use pogo sticks their suicidal thoughts will decease. Because guess what, a lot of kids with GD go thru puberty with no drugs or surgeries and end up feeling better.
And what do you think that demonstrates? A lot of cancer patients go into remission without chemo or amputation, but that says nothing about the efficacy of chemo or amputation when applied with due medical consideration TO THE PEOPLE IT IS DEEMED MEDICALLY BENEFICIAL FOR. The point is that the evidence suggests that, when deemed medically appropriate, it produces positive results. Y'know, like how medicine works.
Go on, then.
Wait what? Are you saying that doctors who don't use GAC tell kids with GD that their identity is a mental disorder? I'd like to evidence of that
Nope, now THAT'S a straw man. Go back and reread what I wrote.
The doctors who have abandoned GAC would instead say to a GD kid something roughly like: "We're taking your feelings of distress seriously. We're here for you. We don't yet know how your sexuality might end up resolving. Maybe you're straight, maybe you're gay, maybe you're trans."
Ah yes, totally a thing a medical professional would say and do, because you say so.