First me secular answer, then my theological one.
Centuries on, I'm guessing probably since the enlightenment at least, of people having aversion to the idea is in my view, the largest factor. In the the past three or so centuries, monarchy based governments fell more and more out of favor, and a not uncommon feature in those governments was incest. Families like the Habsburgs did it to an insane level.
So on top of the visible, very literal insanity it caused a number of rulers (King George III literally once shook the branch of an oak tree believing he was shaking the hand of the king of Prussia), there was also a great disdain for anything associated with monarchy and old ways - so I can imagine that incest would be seen among that.
People do not sway easily. You could similarly talk about something like cannibalism - someone could write in their will (if I die in a car crash, I want you to eat me) and the idea of that sends most people into feeling really sickened, especially family members.
Personally, with my family members, I have lived my whole life seeing them in a way completely divorced from romantic love. I love them yes, but not in any real physical manner. The idea of it makes me feel very uncomfortable.
I'd further say that evolution (or God if you prefer) probably hardcoded into many of us a disdain for being intimate with people too genetically close to us.
Now, into theology. From my perspective, most who practice Judaism, Christianity, or Islam, would probably not move on this matter easily. A lot of Christians especially are extremely committed to "what does the Bible say" and if you veer too far from the intention of the texts, no matter the translation, you are going against God. I think this would be an extremely hard sell, as while I've heard some interesting arguments that homosexuality is not forbidden by the Bible, incest is far more ironclad. Leviticus lists in great detail not to have relations with your parents, children, and numerous family relations.
That would be an line that many would point to. I'm sure others will have other arguments why religion would prevent people from accepting incest, be it tradition, it being forbidden by religious authorities, etc - that I think is rather obvious of an answer.
So I will share a more specific, different kind of one. I'm a member of The Church of Jesus Christ of Latter-day Saints. To us, marriage is one of the most important decisions we undertake in this life (or the next life if there are extraneous circumstances). We believe marriage is not "till death do you part" but in fact lasts forever. Pending faithfulness on both sides, a husband and wife will be united, or sealed to each other, and any children they have for eternity.
Their children, then, upon marrying, weld a new link in this great chain - the husband and wife sealed together, the children sealed to their parents, and then those children being sealed to their own spouses. We believe that this is one of the greatest things beyond this life, that ultimate joy can only be found through the bonds of family.
Incest however, fouls this up considerably. One way to see marriage, I would say, is an act of finding someone out there, and inviting them to share in the joy my family has. In some cases it may even have saving power. Maybe their family is abusive, has problems with alcohol, or something like that. Reaching out to someone then outside of people you've known your whole life, and making them part of your life, thus can bring joy to them, and those around them (not saying necessarily that you can't marry a childhood friend or something.)
And of course, I acknowledge my aversion to incest from just longstanding western culture. Incest sullies the purity in that family chain. Of course, no family line is going to be "perfect" if one could even call it that. Divorces, fallings out between parents and children - all that happens.
But incest... incest to me feels like taking it to a whole other level. Last personal thought - it feels weird to me, because, most of the time, when two people marry, it is after they have known each other for say, maybe a few years, maybe through high school or as far back as junior high.
The thought of siblings having relations just feels disgusting to me, because if that were normalized, you'd watch two brothers at like age five wrestling with each other or something... and some might think about later on...
This in the end will be the big thing standing in the way of this. Many people who align with conservative views (on a large spectrum of stuff, not conservative in the political sense) will take great pause at all this. Normalizing incest to many would just feel extremely awful. It would confirm many people's existing feelings that society is trying to normalize too many behaviors and will not stop, even with such awful things as pedophillia. Whether this is a correct view or not is not for me to say, and I honestly can't say for sure much of the time.
I give all this, as a wide breadth of why people would push against this. I hope especially that my personal background doesnt' come across too much as opinion based - I simply wanted to provide a unique perspective on this.