There are a lot of possible objections to canon Rails Off, but in my mind the cousin angle isn’t one of them, or at least nowhere near a leading reason. I mean would it spell the death of the ship as a canon possibility? Absolutely, since Star Wars is an American production and the taboo against cousin marriage in the States is… frankly a lot stronger than I thought it would be, and a lot stronger than I believe is reasonable.
I mean let’s get this straight, cousin marriage is absolutely forbidden in my culture going back about 5-600 years. Prior to that there are records of marrige between first cousins, aunts and nephews and so on, but the custom disappeared and is now culturally taboo. Koreans consider cousin marriage to be equivalent to sibling marriage and I have internalized that prohibition. The thought of being attracted to any of my cousins makes me ill, personally.
That said, my own feelings about cousin marriage and my culture’s prohibition against it don’t mean the custom is objectively wrong. If there are close family relations and power dynamics involved it can definitely be harmful, but so can relationships between unrelated people. There could be an elevated risk for genetic disorders in offspring, but this is heavily dependent on the population involved. Repeated first-cousin marriage in a small gene pool that carries known genetic disorders is likely to be a bad idea. However, the occasional pair of first cousins, or more distantly related individuals, having children in a reasonably genetically diverse population is unlikely to significantly raise the risk of harm. (I actually first learned this in junior high when I set out to find proof that cousin marriage is disgusting and terrible. Damn you, facts, not confirming my biases and changing my opinion.)
So I absolutely support the rights and validity of first-cousin relationships between consenting adults. I would not do it myself, such a marriage is legally invalid in my country, and the thought is unpleasant to me, but all of that is on me and not on anyone who is in a first-cousin romantic relationship or whose culture allows it.
But yes, due to the taboo in modern America, if FinnSky and ReySol are both canon then Finnrey as a canon romance is as dead as canon Rello. I think Finn and Rey’s relationship would work just as well as a close cousin relationship, and if fans choose to ship them they can be my guest. It would be an unworthy bait-and-switch in my opinion after the romantic subtext in TFA, but that’s this franchise for you.
I just hate the fact that Finnrey shippers in a FinnSky/ReySol universe would get racist shit thrown at them under cover of anti-incest concern, which actually closely parallels the way other cultures are stigmatized by the anti-cousin marriage taboo.