The groups never go extinct. But group selection happens when groups are selected against. The math used to argue against group selection assumes from the outset that group selection does not occur. (This is also true of Maynard Smith's famous haystack model.)

That argument is invalid. Adaptations arise as a result of differential reproductive success. Some haystacks do indeed do better than others in contributing to future haystacks - since they contain more individuals which contribute to the big pool of individuals, from which the next generation of haystacks is produced. So: Maynard Smith's model is just fine in this respect.

The Harpending and Rogers model from 1987 that you critique works in the same way.

Selection does not require different rates of extinction if there are different levels of reproductive sucecss.