Prerequisites for adopting Chinese characters as a writing system?

At one point Vietnamese, Korean and Japanese all used Chinese characters to write their languages. Now only Japanese is does while the others have abandoned “chu nom” and “hanja”. I noticed that these three languages all have 40% or more Chinese vocabulary. I am curious why Mongolian, Tibetan and Manchu never adopted Chinese characters and I am I correct to assume that this is because these languages have far less Chinese vocabulary?

I also think that adopting Chinese characters requires bilingualism in Chinese. In medieval times the literate elite of Vietnam, Japan and Korea all communicated in Classical Chinese. By comparison adopting the Latin alphabet doesn’t require any knowledge of classical Latin. Is this analysis correct?