DISCLAIMER:This joke is homophobic. "Trump being embarrassed for people thinking he had a gay relationship makes him homophobic. Affirming that embarrassment by making it the butt of the joke makes the joke homophobic"
this article explains it well: https://newhumanist.org.uk/articles/5082/the-dangerous-stereotype-of-the-self-hating-homophobe
The implication is that it would be embarrassing for him to have had a gay relationship.
Trump not wanting people to think that he's gay is the other half of the homophobia, and by making that the butt of the joke, we've affirmed it.
I think I can see where you're coming from in that way.
I'm not gay so I'm not as sensitive to these jokes. I am Asian, however, and I dislike "Uncle Roger" for what I think is similar to this. The butt of the joke is that Asian people speak funny. So I think it's similar to what you're saying. The butt of the joke is that it's embarrassing to have a gay relationship.