I had a long post written, and it was pretty good, but then I thought that I would cut this down to this:
1. Your condescending, paternalistic, offensive tone does your effort harm. I'm guessing you're early 20s, if that, and have never been outside an academic environment. I'm not going to say "clearly" or "obviously", because, in my experience, people that say same are talking about things that are not "clear" or "obvious".
2. Name ONE "unique" health issue to the LGBTQ community. ONE. HIV/AIDS? No. Anal cancer? No. Depression? No. Eating disorders? No. Body dysmorphic disorders? No. Heart disease? No. None are "unique", but are more highly represented (and that is from the 300+ page report you linked).
3. You bandy "cultural competence", and that's funny, because I deal with many cultures in my practice, and the staff gets a laugh at how "culturally aware" I am. I deal with LGBT people every day - either colleagues (such as the lesbian couple that works with me - neither is "butch" nor "lipstick lesbian") or patients, who rarely come in for their mental illness, but do for their ankle twists, and productive coughs, and finger abscesses. And do you even know about the "fa'afafine" and "mahu"? I'll bet that you know little or nothing at all. However, I see these folks daily. And they love me, because I treat them like people, and love them for their culture, and who they are.
To hew to the OP, what residency directors want is someone who will complete the program, and not make waves. I know of one gay PD, but he is not "flaming". It's just who he is. If you equate "not being a homosexual stereotype" with "trying to pass as straight", well, all I can do is wish you luck, because, if you make it plainly known about your sexuality, that is going to turn off a LOT of people, including those in influential positions. But you do what you want to do. Sincerely, good luck!