For obvious reasons, this was practically impossible under the draft (which would become a dead letter if "hey I'm gay" was an instant Get Out Of Army Free card).
You really don’t know much about the screening that took place during the draft is my guess, I don’t imagine you were deeply involved in it as I was for about two years.
Remember that the draft ran continuously for about 32 years, WWII had been overwhelmingly a draft war, with 93% of the WWII Army being draftees, along with a high percentage of Marines and Navy. Vietnam was primarily volunteers.
The draft boards and shrinks and medical examiners were pretty hip to the process of separating actual homosexuals from obvious fakers, not a lot of young men of that era could pull off convincing a shrink they were homosexuals, or wanted to.
It wasn’t impossible, it was easier in most cases than determining an honest conscientious objector which was a more common ploy, but no medical exam was going to find anything there, and even a shrink had trouble with that, but looking into their background helped, signing federal documents under oath helped.