The reality is that “social mobility” is a term of art in economics that measures something few ordinary people care about. People mostly care about economic mobility, which is strong in the US. People want the to be better off financially, not more socially mobile. These are not the same thing. You can readily increase your earnings by large amounts without being socially mobile.
The notion that people prefer to be “socially mobile” but poorer isn’t backed by a lot of evidence. People like more money, as a general rule.
Appealing to “social mobility” is something people with money do.