I think you hit the first big issue… if the dogs brought in are very different than what the ideal is, we won't use them much. But looking at the 1/4 dogs, the ones I see look like a Basenji to me and my untrained eye wouldn't know if not told. That said, with breeding coefficients, yeah even 1/8th African does make a difference in the total amt of different genes. And considering ..
Okay lets go to a breed I know well. Rottweilers. Huge amt of dogs compared to Basenjis, right? However, you pick up any 1000 pedigrees, even those foreign born/registered, and in reality the gene pool is not so big. We have an ever increasing problem with heart issues, bone cancer and now diabetes. You can't find lines without those in them. Well you can find people who claim it, but there is no testing to back them or there is testing to prove they lie.
So really I support that the more genes, the healthier any breed. Obviously if we bring dogs that aren't used, it won't help. But then again, if they are overused, it also won't help-- not that it is likely to be an issue.
How unhealthy is the breed? I don't know. You breeders may know. But one positive point is the growth in DNA testing may one day allow us to test them all so that we don't throw the baby out with the bathwater and simply breed carriers with recessive disorders to clear and be fine with things like Fanconi. (Back to Rotties... before we removed long coats from the pool, no matter how spectacular they were otherwise, over a look not related to health. NOW we have a test, a do it yourself at home one in fact!, to test for the gene. Voila, we can keep them in the pool through selective breeding.) But who knows what will rear it's head... just as diabetes is with Rotties and seemingly CUPS is growing in Basenjis. Sorry, I am not for the elimination of breeds, but I am for expanding the gene pool in whatever ways can be healthily done without sacrificing the basics of the breed.
I find it a bit interesting that you have a breed such as this that wasn't originally developed by someone's kennel, yet we act like they MUST be thus and so. I mean, look at my reaction to the African pics posted. I am equally guilty in my "what they should be like." It isn't the new genes that had me salivating over Dr Jo's puppy, it was the utter basenjiness of it.