A citation should make reader's job easier! A textbook with the best exposition, or a survey with all the relevant references (if it exists) is a preferred choice, especially where the original sources are obscure, hard to locate, and/or read. So I would write, "by a theorem of Perelman (see e.g. [Morgan-Tian, Theorem 1.2.3])".

Speaking about promotion, by the time the stuff that made it into textbooks, this is (usually) no longer an issue, and in many cases the person responsible for the original result may have achieved a god-like status. Then it is common to even omit the author's name, let alone give a reference to the original source, e.g. people say "by the s-cobordism theorem".