Neither goats or sheep are native to Asia (at least the species that we domesticated; there may be some wild versions I am unaware of). Animal and plant domestication in China (which later spread to the rest of East Asia) is a separate event than domestication in the Near East (where we get our sheep and goats of today from). So, possibly, (even though of course by the time periods you're talking about there would have been trade between East Asia and the Near East) sheep/goats just never became that popular in East Asia. My other guess would be that rice is the main food plant in East Asia, as opposed to wheat and similar grains in the Near East and Europe, and sheep/goats may not do as well on rice.