If the group with the trait has on average a slightly higher reproduction rate than a group without, that is sufficient.

This doesn't mean that any particular individual will have a higher rate (although most will).

One hypothesis for the spread of Christianity in the Roman Empire is that Christianity provided social services that improved reproductive rates by a couple of percentage points. Maintain that over a particularly stressful upheaval in the social order and you don't need many generations to significantly outbreed your rivals.