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“If you have a child with your cousin the likelihood is there’ll be a genetic problem.”

Sorry, but that's nonsense.

It implies a greater than 50% chance of genetic defects.

Actual chance for first-generation cousin marriage is going from 2 to 3% chance to 4 to 6% chance.

I'm not sure of your percentages, but the concept is reasonably accurate. I'm an amateur genealogist and can tell you the first cousin marriages in early America, especially in isolated areas was not at all uncommon, possibly as much as 10% in some locales. Ditto for the old country.

This isn't because the people were weird or anything of the sort, simply because they didn't get around much and the pool of available marriage partners were limited.

If you think of a random first cousin marriage like a pair of dice, your chances of rolling snake eyes in one in 36. However, the offspring produced will include die with maybe two ones on it. Roll those together and the chances go up from 1 in 36 to 1 in 9. They produce die with three ones. Roll two of those together and your chances of a genetic defect go from 1 in 9 to 1 in 4. Those are the type of increasing odds you play with repeating first cousin marriages generation after generation. Not exactly the same odds, of course, but the same concept.

My wife and I discovered we are also distant cousins through different children of a Rev. John Crandall (1618-1676), a contemporary of Roger Williams, founder of the Colony of Rhode Island. Almost any couple in America who has ancestors who were here or arrived before 1670 (about the time the United Kingdom starting sending undesirables to America in large numbers) will have a similar, if not even closer, cousin relationship whether they can document it or not.