Second, I don't know of any treatments for gender dysphoria (in trans people, not people with other conditions that've been misdiagnosed) that "fix" the mind and actually work. Gender reassignment, on the other hand, works, in the sense that it improves peoples' lives (see https://www.scimex.org/newsfeed/transgender-teens-receiving-... for example).
Note that not all transgender people experience gender dysphoria. Some are fine with the bodies they were born with, and for them gender reassignment would be unnecessary and irrelevant.
Also, note that this is different from treatment for e.g. anorexia nervosa. If someone with anorexia loses weight, it doesn't help; they'll continue to see themselves as overweight. Treatment for anorexia has to focus on the patient's mind. Helping them lose weight would make their outcome worse, not better, which is why it's not done.
Scott Alexander gives an interesting analogy to a case from the mental hospital where he works here: https://slatestarcodex.com/2014/11/21/the-categories-were-ma.... A woman with OCD was constantly worrying that she'd left her hair dryer on, and was having to drive home 10-20 times a day to check whether it was really off, or was on and going to burn the house down. Nothing they'd tried worked, until someone suggested she take the hair dryer with her. That effectively solved the problem for her, because she could always just look over at the dryer, see that it was unplugged, and go about her business. But caused a huge controversy among the psychiatrists between those who thought "This Is Not How One Treats Obsessive Compulsive Disorder" vs those who just said "it worked".
I'm with the "do what works" crowd.