Being transgender is not considered a mental illness. Nor is does being diagnosed as transgender result in a 'nearly automatic' diagnosis of schizophrenia. There is an _increased rate_ of schizophrenia among transgender people. But it still remains uncommon - less than 1 to 5% of transgender people are _also_ diagnosed as schizophrenic.
Answers to Your Questions About Transgender People, Gender Identity and Gender Expression from the American Psychological Association
Is being transgender a mental disorder?
A psychological state is considered a mental disorder only if it causes significant distress or disability. Many transgender people do not experience their gender as distressing or disabling, which implies that identifying as transgender does not constitute a mental disorder. For these individuals, the significant problem is finding affordable resources, such as counseling, hormone therapy, medical procedures and the social support necessary to freely express their gender identity and minimize discrimination. Many other obstacles may lead to distress, including a lack of acceptance within society, direct or indirect experiences with discrimination, or assault. These experiences may lead many transgender people to suffer with anxiety, depression or related disorders at higher rates than nontransgender persons.
According to the Diagnostic and Statistical Manual of Mental Disorders (DSM-5), people who experience intense, persistent gender incongruence can be given the diagnosis of "gender dysphoria." Some contend that the diagnosis inappropriately pathologizes gender noncongruence and should be eliminated. Others argue that it is essential to retain the diagnosis to ensure access to care. The International Classification of Diseases (ICD) is under revision and there may be changes to its current classification of intense persistent gender incongruence as "gender identity disorder."