Build Back Better and other policy measures undertaken by the Biden administration were carefully calibrated to try to help these very groups. Those disadvantaged by globalization and skill-biased technological change. Your contention does not square with reality. What the Democrats were not willing to do was to scapegoat certain groups like trans people and immigrants. That proved to be a far more effective way to win over those voters.
What you’re describing is policy, not messaging. It is not my belief that Dems must win the culture war to win elections, nor is it that they must abandon marginalized groups to do so. My main contention is that you can win with a majority of voters who disagree with you on social issues if you just learn how to talk to them about how you might be able to help them.
Republicans are in the middle of their own self-implosion on this same front. I’ve said it before and I’ll say it again, MAGA has drastically overestimated their relative strength of certain aspects of the culture war and have subsequently overplayed their hand in that regard. People will get on board with keeping biological males out of women’s sports and other divides between the left and the right, but people are not largely hateful. The casual cruelty the current Administration and their friends in Congress are employing on immigration and gender will backfire, just as calling voters with little exposure to other cultures in their daily lives racist or homophobic backfired on the Dems.