“The right to execute” is not an issue…except in Muslim countries… so unless you live in such a place, not seeing it.
As far as you trying to understand the objections of others, I don’t see that in your post, nor why it matters. It it seems to me that a certain segment of our population are keen at operating within shades of grey, which is messy… and by that I mean, if one can come up with a justification for a thing, then society should act on it. Perhaps that is why you have an interest in a “reason” that might make it rational to clamp down on gays… (like students demanding free tuition for “blacks”, because slavery). Any reason will do. (Not that you have this as a desired outcome… only that you would believe it matters)
I think this is the wrong approach. The idea ought to be that people are free… life, liberty, and the pursuit of whatever… until you would violate others.
In such a structure there are no gay people. There is no basis for laws against gay “sex”. You need not concern yourself about “why” someone would object to it, because it doesn’t matter.
But now we come face to face with your claim that it is wrong to treat people attracted to the same sex, differently from those attracted to the opposite… which is that just like everything else: we are free to do exactly that. It’s not wrong. It’s just a choice.
For example, since homosexuality is literally abnormal, one might take the clinical view that the abnormality might be one day understood and treated or prevented outright.
Imagine a “vaccine” for women or men to use before they have children that prevents the abnormality (provided it is genetically based).
If a gay is a “type” of person (like a sub species) then it might be considered genocide for such a vaccine to be used, on the other hand, if it’s just a proclivity that can be medically treated, then it is obviously not genocide.
So these issues are not trivial, nor is it clear that it is wrong to treat such persons differently, in the same way we treat people with gambling addiction differently, suicidal tendencies, Star Trek obsession and pedophiles differently.