There's certainly lots of anecdotal evidence that many people's brains just seem to "work better" once on the correct hormones, and generally speaking transition has very good outcomes - people who have transitioned are generally doing better on key indicators, though still not as well as cis people. The people behind the studies believe that this might be down to how prejudice and discrimination affects trans people even after transition, though there's no further studies in that direction yet. (For a comparison, I believe gay people were doing worse than the general population on the same key indicators before general acceptance, and it's been gradually getting better.)
Here's a question, though: if we're not who we are in our own minds, who are we? Are you and I really only what others can see?