The "right side of history" arugment is always flawed. It depends on this idea that "progress" is inevitable. It's not empirically supported and the philosophical basis for believing it is rarely established by those using the term.
What people really mean is that they think that is the direction of travel in society and they don't want to be found espousing a minority opinion. Which is a little cowardly when it comes down to it.
As far the gay rights thing - part of the reason this is compelling to people is that gay rights are often argued on the basis of identity. As soon as we take that approach, it means that other identity based arguments can be hitched to it. So to some extent we've shot ourselves in the foot by being blase about accepting those kinds of arguments, which were never well formed.
But the basic issue is this - whether or not it should be legal, or morally ok, to have sex with whomever you can get to agree to have sex with you (who is an adult etc..) is simply not the same question as whether or not individuals can or should be able to assert anything about their sex or gender and that must be accepted socially and legally.
It doesn't matter if you agree with both ideas, disagree with both, or agree with one but not the other. They are not making the same proposition and so they need to be studied and argued on their own merits pertaining to their own characteristics and contexts.
The same is true for civil rights of blacks, or animal rights, or women's rights, and gay rights. There may be some parallels and overlap, insights that can be passed from one to another, but they are not identical. THat is what a comparison is, looking at different things to see what elements are the same and what aren't. You cannot say that because we should not discriminate on the basis of race, it follows that we should not discriminate on the basis or sexuality, or that what would constitute discrimination in each would be the same. We can't talk about discrimination against women in the same way as discrimination on the basis or race - there are important differences.
It's just a sloppy, sloppy argument.