The population bottleneck was far less dramatic, as it happened over centuries.

When population growth is effectively 0, the size of the male lineage of the average man is approximately 1 in each generation. But random variation means that the male lineages of most men don't survive that many generations.

Now imagine a stratified society, where elite men have more children than commoners and social status is largely hereditary from father to son. The average elite man will have >1 sons surviving into adulthood, while the average commoner will have fewer than one adult son. Repeat that over many generations, and lineages that started from an elite man will likely spread, while lineages starting from a commoner will likely go extinct.