"I want some milk."
Certainly in conversation we are using "ミルク" rather than "牛乳".
But the word "牛乳" is officially defined by the rule. "牛乳" is milk only "cow". etc. So, the milk company prints the word "牛乳" on the package. It is useful when you are looking for milk at the supermarket. And there are few Japanese who do not know "牛乳". The word is still alive.
the problem here is that the English translation contains a verb and a direct object -- I want milk. But that's not actually how the sentence functions in Japanese. 欲しい is an adjective, not a verb. It might be helpful to think of the direct translation as "Milk is desirable [to me]." Therefore, 牛乳 is actually the subject of the sentence, and thus takes the が particle.
From what I've seen, you're thinking along the right track. If i just got home after a long day at work and was tired/hot, I might say "牛乳が欲しいです" meaning that i desire a cool glass of milk to help me relax. If, however, I was sitting at the table with someone who just poured themselves a glass of milk, I'd say "牛乳をください" essentially asking them to pass the milk.
I've always thought of ください and お願いします as "as for that thing you have/near you/you have some influence over, can I get involved with it?" If you see an ad for ice cream, you woudln't use ください to your friend, because they can't give you any, but you would tell them you want some (and maybe invite them to go get some), however, when at the ice cream shop, you'd say ください to the server, since they can influence/facilitate the delivery of ice cream to you.
ください is sort of like asking for it to be given/provided to you, whereas 欲しい is just stating a desire to have. You can express your desire/want of a million dollars, but using ください would get you strange looks, since you're essentially asking them for the million dollars. Think of ください as more of a "please pass the..." phrase.