The same grim idea has occurred to me too. Non-immunes are allowed to breed, but newborns are intentionally exposed to the Rash. In one generation, you have no more non-immune young people.
All true. In the first generation, though, you might not have any problems at all (again depending on starting point), and that may give time for your little community to find another.
...
As those small communities probably don't have access to medical resources to assert immunity, I can easily see the process of exposing newborns to the rash becoming a ritual.
It would have a practical side, allowing the identification of those "too weak" to live, but could also assume a religious meaning: "Let's ask the gods if this child shall live."
I can also see another story here: A couple that decides to flee before birth to avoid having their baby tested.
As for the incest, I see that happening in very small and isolated groups, but the knowledge and the social taboo associeted with it would last for a long time, I believe, and deter any community that had another option, even if it implies some travelling...
As I considered that, I recalled finding that isolated tribes in the Amazonian forest have, deeply embebed into their old traditions (that predate any cultural contact with the rest of the World), an absolute prohibition regarding any marriage or relationship between people from the same tribe.
Those tribes populations are usually 30-50 people, and they live far enough from each other to spend weeks or months without contact with any another tribe.
So they organize meetings between tribes where they exchange people. When they are peaceful, which seems to be most of the time, men and women on the proper age travel, sometimes for days, to attend festivals where they have the chance to meet someone deemed suitable by them
and by their families.
If they are at war they may capture woman from other tribes, but AFAIK they don't go to war for that reason.
If a tribe doesn't have access to any woman from outside for a long time (if they live too far in the forest, or are unnable to attend meetings, or there's some other cultural or religious reason), the young men will depart and move to other tribes, and that tribe will disappear as the remaining members get old and die.
BTW, you're lucky to have parents with such good literary taste,
lwise! I had to resort to some older cousins...