I was draft eligible during Vietnam and can tell you that it was certainly widely believed that claiming to be homosexual would get you out, although I don’t know anyone who tried that. It was also rumored that the military had tired of that dodge and if you tried it, you might not like the unit you wound up in. I found nothing to support that. The military certainly did prosecute for sodomy during WW II and Korea.
Back in college I had noted civil rights leader Julian Bond for a professor on that subject, and he told us that when he had to show up for induction, he was deemed morally unfit, presumably on the basis of his active involvement with the movement.
I always figured that’s what the “Klinger” character represented in MASH. That explains his dressing up as a woman all the time? Sure the show claimed he was trying to get a Section 8 dismissal; but wasn’t it really all about him trying to get booted for being homosexual?