This is a framing issue. You have two separate conclusions: (1) Being transgender "should be treated as a mental disorder" (2) Recognizing trans people as their chosen gender is societally harmful
First, (1): If your mind does not match your body, which one defines "you?" And it's fairly philosophical/ hard to get a clear answer.
I'm not a medical professional, so let's avoid the science side of this discussion by assuming that there is a way to medically induce someone who identifies as trans to accept their biological sex as their gender, and to do so with minimal to no side effects. Is that a good idea?
For a lot of people, even something as simple as taking antidepressants is threatening to their sense of self — i.e. "If the drugs change the way my mind works, that's not 'me' anymore." While most people recognize that isn't a terribly healthy way to view depression, the question becomes more difficult to answer as you consider other more significant treatments aimed at affecting the mind. At what point are you no longer enabling a person to "be themselves" but instead directly altering who they are?
By definition, being trans is about self-identification. Even if you could change the mind to fit the body, treating it as a mental disorder would be changing who they are. Changing the body to fit your self-perception, then, isn't changing who you "are" because that's already how you see yourself.
On to (2): To what extent should society recognize your self-perception? 99% of the time, the answer is "why not?" As you yourself state, men and women are free to dress and act how they want. Does using "she" to refer to Caitlyn Jenner actually harm anyone? No. Does letting her use the women's room pose any risk? No. (For the hyperventilating "what if a rapist puts on a dress and walks in?" crowd, the sign on the door isn't a magical barrier that keeps out people who wear pants. Crimes are crimes, and forcing trans-women into the men's room is quite a bit more dangerous than letting them in the women's.)
To the extent that it does make a difference, there are barriers in place to prevent abuse. A man can't join women's sports teams just by claiming to be trans. Assuming the team/league even allows it, they'll generally require documentation of medical transitioning, and require the person to take the medication to regulate the hormones that create the physical differences you believe pose a danger. It's the inverse of PED testing, and the risk of noncompliance isn't meaningfully different.
But even if that were indisputably insufficient, at best you've demonstrated that trans people should be restricted from participating in contact sports. Why should how we treat contact sports dictate whether I call some random person on the street "he" or "she?" And now we're back to (1). When there's no harm to it, why should my sensibilities get to interfere in how someone identifies themself?