Does that "rule" not apply in Britain as well?

For some accents, yes. (Having a middle-class regional accent, from Edinburgh or south-east England, never hurt anybody). But there's a widespread American belief that class stereotyping (to the extent that it limits social mobility) is something that only happens the other side of the Atlantic.

This particular one is relevant here because so many folksongs have been collected from source singers who have that sort of accent. Over a very long period; Sharp and Karpeles went to the Appalachians because it was home to a marginalized community cut off from the mainstream culture, and the Lomaxes found the same decades later. Looks like not much has changed over a century.