I do not, however, think that the comparison to homophobia or other discrimination against LGBTQ+ people is a good comparison. The majority of relationships that LGBTQ+ people engage in are consensual and do not cause harm to anyone. The majority of cases of incest involve sexual abuse and frequently pedophilia. Offspring of close relatives are at high risk for significant biological and social harm (in cases of abuse add psychological harm).
I think the guy you’re referring to isn’t trying to compare incest to gay rights or anything. He’s merely pointing out that the argument against incest among consentual couples is a slippery slope argument similar to the slippery slope arguments used by the far right to deny the LGBT community their rights: “If we let them do _________, then the next thing they’re gonna want is ___________”. It’s a bad argument to make no matter the subject or which side of an issue you’re on. There are plenty of legitimate reasons not to support incest without having to resort to slippery slope arguments.
There’s also the fact that if one were to seriously try to legalize incest among consenting adults, the immediate response from the right would be “See? We told you that if we started letting gays marry, they’d want to marry their cousins next! What’s next, their pets?”. And you and I both know that they would immediately start using this argument to further isolate and marginalize the LGBT community, even if they try to distance themselves from the idea.
Go back some time and see what happened when NAMBLA tried to shoehorn themselves into the LGBT rights movement. The LGBT community immediately denounced the group and distanced themselves from them as quickly and as forcefully as they could, and the far right still shit all over them for it, saying “See? They’re starting already!”. The same thing would happen here – the LGBT community would distance themselves from the idea right from the get-go, but that wouldn’t stop bigots from blaming them anyway even though they have nothing to do with it.