There is totally a genetic problem caused by multiple generations marrying inside the family though, which is part of where the taboo comes from, isolated families or aristocrats marrying their first cousins for literally centuries.

So I did some digging I and found this: https://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.0060320, which says:

Perhaps surprisingly, these bans are not attributable to the rise of eugenics. Popular assumptions about hereditary risk and an associated need to control reproduction were widespread before the emergence of an organized eugenics movement around the turn of the 20th century. Indeed, most prominent American eugenists were, at best, lukewarm about the laws, which they thought both indiscriminate in their effects and difficult to enforce [2]. In the view of many eugenists, sterilization of the unfit would be a far more effective means of improving the race.

Nonetheless, in both the US and Europe, the frequency of first-cousin marriage—a practice that had often been favored, especially by elites—sharply declined during the second half of the 19th century [3]. (The reasons are both complex and contested, but likely include improved transportation and communication, which increased the range of marriage partners; a decline in family size, which limited the number of marriageable cousins; and greater female mobility and autonomy [4,5].) The fact that no European country barred cousins from marrying, while many US states did and still do, has often been interpreted as proof of a special American animosity toward the practice [6]. But this explanation ignores a number of factors, including the ease with which a handful of highly motivated activists—or even one individual—can be effective in the decentralized American system, especially when feelings do not run high on the other side of an issue. The recent Texas experience, where a state representative quietly tacked an amendment barring first-cousin marriage onto a child protection bill, is a case in point.

The laws must also be viewed in the context of a new, post–Civil War acceptance of the need for state oversight of education, commerce, and health and safety, including marriage and the family. Beginning in the 1860s, many states passed anti-miscegenation laws, increased the statutory age of marriage, and adopted or expanded medical and mental-capacity restrictions in marriage law [7]. **Thus, laws prohibiting cousin marriage were but one aspect of a more general trend to broaden state authority in areas previously considered private. And unlike the situation in Britain and much of Europe, cousin marriage in the US was associated not with the aristocracy and upper middle class but with much easier targets: immigrants and the rural poor. **In any case, by the late nineteenth century, in Europe as well as the US, marrying one's cousin had come to be viewed as reckless, and today, despite its continued popularity in many societies and among European elites historically, the practice is highly stigmatized in the West (and parts of Asia—the People's Republic of China, Taiwan, and both North and South Korea also prohibit cousin marriage) [8–11].

So it doesn't seem like the rise in those laws and the associated taboo really had anything to do with genetic disorders, but rather other factors.