Ignoring a good story is dangerous: We can qualify what constitutes a 'good story' - but insisting that stories be ignored and only statistics be looked at is quite extreme. As you say:
"What do these stories miss? They miss that reality is messy and complicated. These stories miss, perhaps, that only a small fraction of beneficiaries experienced these benefits, or that the program only worked in certain contexts, or that not all handsewn decoration producers are equally effective entrepreneurs"
But this is just bad story-telling. This is the story-telling of a politician or an ideologue and not that of a rigorous researcher. And to be honest, these stories probably do as badly as talking only about statistical average treatment effects.
Any researcher who dismisses stories is being too harsh and if stories = qualitative data, being this dismissive about them just demonstrates a lack of rigour. So let's not make decisions on stories alone. Let's also please not make decisions on statistics alone