The Tampa Humane Society became a "no kill" shelter a couple of years ago, so all strays, etc now go to Animal Control. I have been called by the Humane Society (gave them my name and number before BRAT existed) to pull several basenjis, one turned out to be a really cute mix, over the last 6 years, but for them rescue is a last resort, they prefer to adopt dogs out themselves. In the past, Animal Control has called rescue groups for all purebreed dogs, but even so, I don't doubt that basenjis have fallen through the cracks, or were deemed unadoptable (bite history, aggression, injured).
But this quote is from a Hillsborough County Animal Control person about the St. Pete SPCA, who also does work with rescue. Again, it could be true, but we are talking about 3 different facilities, St. Pete SPCA, Tampa Humane Society, and Animal Control, all of whom euthanized till recently.