It doesn't really surprise me much that lactase persistence should be so strongly selected for in a relatively short period of time. Cows and other ungulates possess a rather magical ability to turn something that can't be digested by humans (cellulose) into something that can (milk). Opening up the huge caloric and nutrient resource of cellulose to humans in a relatively replenishable way had to be a giant benefit t…
Lactase persistence isn't the only (or even first) way to convert inedible flora into food. For one, you can simply eat the cows, the way it was done for most of human history. If you absolutely insist on dairy though, the genes for lactase persistence post-date dairying by a few millennia and many people who rely primarily on dairy today still lack the genes, like most Mongolians. They've adapted different productio…
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.