Besides what sibling comments already mentioned, eating the livestock directly meant having to herd larger groups of animals. One hypothesis that comes to mind that explains how lactase persistence might spread so fast was the enabling auto-sufficiency on small family-level herding that helped with survival. The primary target of plagues and war were the large settlements, and I imagine it was very useful to be able to live on small and thus low profile herd that roamed away in the wilderness, far away from the population-limiting events, or at least sitting them out somewhere close by but safe. This is harder to do with large herds, which for one consume the grassing stock faster, require more active movement, and thus confers inferior stealthy means of existence that is so important in face of wars of that time.