A lot of what you said is reasonable, but I think you're really stretching your argument when you say that 1) liberals refrain from advocating for the "two-parent family" because to do so would imply pushing back on workplace equity and benefits for single parents, and 2) support for fathers' inclusion in paid family leave is proof that liberals care about two-parent families. I think you are seriously underplaying a very real strain in left-wing thought, which views the "traditional family" with extreme suspicion, and associates exhortations for couples to marry before they have kids, to stay together for the sake of the kids, and so on, as essentially patriarchal, religious encroachments on personal liberty.

Since you want hard poll evidence, I browsed Gallup and found a poll:

41% of Republicans vs. 61% of Democrats think having kids outside of marriage is "morally acceptable". [1]

61% of Republicans vs. 73% of Democrats think divorce is acceptable. [1]

Interestingly, you can look at [2] and see that these issues fall pretty neatly into OP's argument about "luxury beliefs". For instance, on the "having kids outside of marriage" issue, only 38% of black Democrats think it is morally acceptable, but 64% of non-black Democrats think it is morally acceptable. The black vs. non-black numbers for sex outside of marriage are 46% to 68%. Given the prevalence of exactly these behaviors in the black community relative to the white community, I think that this constitutes pretty hard evidence that OP's "luxury beliefs" hypothesis has a lot going for it.

[1] https://news.gallup.com/poll/137357/Four-Moral-Issues-Sharpl...

[2] https://news.gallup.com/poll/112807/Blacks-Conservative-Repu...