Since nobody knows who their daddy is, we're going to have LOTS of half-brothers and half-sisters inadvertently marrying each other. Especially since you have these jokers who have hundreds of sperm-donor offspring all over the country.
This has already become a problem in the dog world. AKC allows AI breeding, and it's certainly convenient and less risky for sire and dam, BUT there's a downside. Because a sire can father hundreds and hundreds of puppies via AI, many more than he can through live cover, when a dog becomes very popular (as a show winner, or a field trial winner, or whatever), not only does everybody want to breed to him, everybody CAN. There was an admittedly stellar field trial dog - 2XNAFC 2XCNAFC FC CFC EbonStar Lean Mac - won the National Field trial twice, and was very "prepotent" i.e. passed his winning characteristics on to his offspring. So everybody had puppies by him. Now, you're hard put to find a pedigree that doesn't have him in it (often multiple times), which has become a problem because he seems to have carried a couple of genetic faults along with his winning ways . . . so now people are advertising stud dogs in the retriever magazines as "NO LEAN MAC" . . . !
To apply this thinking to humans is stupidity on a colossal scale, for lots of reasons.
Wasn’t it also you who told me about modern Siamese cats having their basic temperament changed by one elegant but ill tempered tom some years ago? Very popular because he was typey, and his nasty nature bred true as well?