>but reporting that this is what parents often tell their kids in private //
I correct my childrens' pronunciations. I've honestly never thought of it as a class thing. In part it's "inherited", so it could have consciously been a class thing for one of my ancestors. But really to me it just seems necessary for preliterate speakers to understand the proper way to say a word based on its spelling, which they don't know. I find myself correcting USA-ian word use ("garbage") and pronunciation far more with my youngest than I had to at that age with my oldest ... we probably let him watch too much TV (several shows, like Paw Patrol have shifted to USA versions from British English versions).
It's hard to discern every phoneme from a new word, and hard to say some of them.
I do shift my accent, and vocabulary, to mark myself as a local I guess (when I'm back in that area of the country). If I get to use certain words in their localised (to about a 20mi long area) meaning it makes me happy for some reason. But I've never consciously worked on my accent to sound more/less posh; but have modified it to be more understood.
My wife is what I call a 'sympathetic speaker', she very quickly adopts the accent of those she's speaking with. An interesting phenomenon.