https://twitter.com/BernieSanders/status/798192678785716224
Bernie grew up in Jewish Brooklyn in the 40s and 50s, so he wouldn't have been seen as a member of the generic white American working class (though he does represent this demographic as a congressman and senator from Vermont).
Also, I really don't think one can appeal specifically and explicitly to the WWC in a progressive manner.
You would not have said this as recently as ten years ago. The issue is that the character of the "WWC" has changed over the past few decades. What used to be widely accepted in these communities has changed with the destruction of organized labor and the death of the "Greatest Generation" (and the Silent Generation and many Baby Boomers; those born later than ~1960-1965 or so won't have much of a memory of organized labor or the left). Also worth noting that what is considered to be "progressive" has changed as well.