India uses milk mostly in converted forms that are low lactose.
Italy didn't have the population/resource stress that China did. Sure, China could have pursued dairying, but they didn't. I've not seen careful studies that indicate the specifics scientifically, but the reasons usually given are as I indicated:
Beef is resource intensive.
Arable land is scarce compared to the size of the population. This is often tied to the rise of the wok techniques as well which are high fuel efficiency and short cooking times.
The Chinese cookbooks I have from the 70s and up to the mid 80s have very little beef in them. The beef dishes are all Northern China. They rely more on sugar in their marination than rice wine. Now, it's fairly common to use sugar instead of rice in Chinese marinades, but the other interesting part of that is this. Rice is a Southern China crop for the most part and wheat is a Northern China crop for the most part. You can see this split in the cuisine here with beef (and lamb somewhat too). The cow was a good plow animal for wheat fields, but not so good for a rice paddy perhaps?
These beef dishes were based around older tougher animals and it is this heritage that gave rise to using baking soda to "tenderize" the beef. From the mid-80s forward, beef becomes more common and the recipes are written to suggest that the beef is of a higher quality and more tender. Wine use in the marination increases and sugar decreases. Reflected in the cookbooks is the start of a change in the food culture of China.
Also, the Chinese have for the most part killed and cooked their meat fresh for the meal. Fish and chickens were bought live and killed at home. Pork, being larger was an exception and this was usually done in a shop that had the oven for roasting and handling the cuts of pork. Cows are larger still and present more issues of scale.
http://www.csu.edu.au/__data/assets/pdf_file/0010/207694/CN-beef.pdf is a study into Chinese Beef consumption. Consumption was pretty low until about 1990 (when markets opened and incomes rose). Price and difficulty of cooking are the two top hurdles to beef acceptance. Disliking smell and flavor were numbers 3 and 4.
It seems they didn't find beef that rewarding as food animals to concentrate enough of them together to develop a dairy culture and reward lactose tolerance.