I think cheese doesn't appear in American style chinese food more because of the way that american style chinese food has evolved (or hasn't evolved) as a cuisine, rather than because milk and cheese aren't traditional chinese ingredients.
I am a Chinese american but also lived in Hong Kong and Taiwan and traveled in China and Singapore, so I'm familiar with americanized Chinese food as well as Chinese food in Hong Kong, Taiwan and China. Food in China, Taiwan and Hong Kong has evolved and changed in really complicated ways as they modernized and got access to new ingredients and exposure to western cultures like Britain and other asian cultures like Japan.
People in China, Hong Kong and Taiwan definitely consume dairy products in spite of whatever lactose intolerance they might have. Milk and yogurt are commonly consumed in China today. Milk appears in certain traditional Chinese desserts (almond tofu jelly and steamed milk). Parmesan and even mozzarella cheese is used as a topping for 'baked rice' in Hong Kong, which are traditional casserole dishes that probably developed with some British influence. Fluffy baked cheese cake is a popular dessert in Hong Kong and Taiwan bakeries (probably result of Japanese bakery influence). Milk is very popular in beverages like milk tea, yin-yeung (mixture of coffee and tea), bubble tea, and coffee.
While food in China, Hong Kong and Taiwan has evolved to incorporate nontraditional ingredients such as milk and cheese gradually, it is curious that this hasn't happened with american style chinese food despite the availability of milk and cheese products.