Why do we need У in "Ножка у сту́ла сломалась"?

Question

In the sentence "Ножка у сту́ла сломалась" (the chair leg broke), why do we need the у? The genitive case on "стул" already lets you know that it's the *chair's* leg that's broken. More generally, when and why is у used to indicate posession in Russian rather than simply using the genitive case? By the way, I'm aware that "сту́ла" is already in the genitive, I'm asking why у is there as well.

Answer 1

 One of the meanings of the preposition "Y" is "Included accessory" (appurtenance), for example, "крыша у нашего дома зелёная, а у наших соседей коричневая", and it is similar to using Genitive case "ножка стула" or "крыша дома". Both situations expressing possession/belonging to something as a part require a Genitive case, as well as after the preposition "y" - with any meaning of the sentence, the Genitive case is a must.

Answer 2

У is supposed to be where regardless of any rules since the basic form of the word is "стУл".