It has already been researched, for decades; it’s just that the findings of the research don’t match up with the image that the autogynephile males want to project to the world, so it gets suppressed.
Here is an article by Anne Lawrence that goes into some of the issues you mentioned:
Here’s an article about what a bunch of angry males in dresses did to J. Michael Bailey, a psychologist who dared to publish a book accurately describing autogynephile behavior:
“…dissatisfied with the option of merely criticizing the book, a small number of transgender activists worked to try to ruin Bailey professionally and personally. Largely under the leadership of three prominent transwomen—Lynn Conway (a world-renowned computer scientist at the University of Michigan), Andrea James (a Hollywood-based trans-consumer advocate and an entrepreneurial consultant on trans issues), and Deirdre McCloskey (a Distinguished Professor of Economics, History, English, and Communication at the University of Illinois at Chicago)—they organized charges of scientific misconduct against Bailey, including charges that he lacked informed consent from research subjects, that he failed to obtain Institutional Review Board (IRB) permission for human subjects research, and that he had sexual relations with a transsexual research subject. They successfully pushed for a top-level investigation of these charges at Northwestern University and for numerous press reports about Bailey’s alleged misdeeds. They successfully arranged a protest against the book’s nomination for a Lambda Literary Foundation (LLF) award and tried to get Bailey’s colleagues (including his closest departmental colleagues) to turn against him or at least distance themselves from him. They devoted elaborate Websites to criticizing and mocking him and his book and anyone with any positive relationship with him. One activist in particular, namely Andrea James, also used the Web to publicly harass Bailey’s children, his ex-wife, his girlfriend, and his friends.”
People don’t research/publish anything except glowing trans-positive stuff anymore because they don’t want their lives ruined and their children threatened. They probably couldn’t get funding anymore, regardless, because the people at the institutions providing the funding also don’t want their lives ruined.