^^That's if they even knew what caused it. When I first started getting some gastrointestinal problems in my thirties if I ate too much cheese or if I ate a lot of ice cream, I had no idea that I was now slightly lactose intolerant, or perhaps intolerant of the milk protein. (Btw, I carry two copies of the lactase persistence gene.) The doctor had to have me eliminate one food after another, and that was after lots of tests for more serious diseases.
Just to clarify, though, there are a lot of goat, sheep and cow bones in Neolithic refuse pits in Anatolia as well as the Balkans and central Europe. Maybe they only butchered the animals when they got old? The ratio of which animals were consumed depended on the terrain. They almost always also hunted. On the other hand, there aren't a lot of remains of these dairy products, far less than there are of grains, pulses or even meat, so I doubt they consumed huge quantities of them.
Nor am I convinced they would necessarily have fed a lot of milk in its liquid form to their young children.
For example, in comparison with northern European populations, we have in Liguria/alta Toscana, my mother's area, a lot more people who are lactose intolerant, or, at least, they don't have the appropriate gene, or only have one copy. I don't know if that's the reason, but although it may be different now since Italians have taken on some "foreign" habits, once children were weaned the first foods were broths, bread soaked in broth, pastina with a little bit of butter and so on. A little older and you did get caffe e latte, or a bowl of milk with a little espresso in it, sugar, and again, soaked bread. Later yet, cappuccino, which has a lot of milk. However, I remember being told you shouldn't have milk with "real" meals, i.e. lunch or dinner, as it would spoil your digestion. Btw, there's a peasant saying: pan e vin fan un bel fantin. Or, bread and wine make a beautiful baby. No mention of milk.
Warm milk and honey before bed was something my mother occasionally did, but my father never.
All of that said, even as adults, our sofrito always starts with butter as well as oil, we always put butter on bread in the morning, we eat a lot of pecorino cheese as well as parmigiano, we often dress ravioli with butter and sage sauce, and I never heard anyone complain. Maybe in moderation there's no problem. Nobody is drinking a liter of milk.
Things are a bit different in the mountains of Emilia. Butter and cream sauces often saturate all the pastas, a fact which even Italians often seem to not know, with their horror of things like the Americanized fettuccine alfredo and claims that Italians would never put cream on pasta. My father's people eat even more cheese than Ligurians it seems to me. It makes sense as I think there may be more cows than people up there.
My father told me that as a child, when they went to visit their grandparents, they would drink the milk right from the cow, but even there I never, ever, saw anyone drink a glass of milk with a meal, for example. I was shocked by it when I saw American families doing that. Even today, the only time I would ever drink some milk would be after dunking some oreos in it, or in a hot toddy of some sort. My brother, on the other hand, got used to it through his friends, so we kept it for him, but never at the table with food.