I agree with the angle you're coming from by I think you miss the bullseye.
This is true. However, hard-left outlets and the media are forever normalizing all kinds of sexual behaviors. And- look- you can't have a culture of promiscuous sex and keep all children born in wedlock. Sex makes babies- especially for poor women who are less likely to abort in order to pursue an education or establish their careers.
Take this just a little bit further. Why would normalization be a problem to red tribe?
Mostly (I think) because it removes the ability of society to enforce the norms themselves via social pressure/shaming/penalties for aberration. In the absence of legal penalties for certain behaviors, that was the main method for pushing people towards the 'ideal' goal of family formation.
And when these pressures are gone, it is not particularly surprising that the goal they were pushing towards (creating and staying as a family) is being thwarted as well.
For instance, liberals largely preach the following:
Having sex outside of marriage is fine. Even if you're a teenager. In fact it should be encouraged. Shaming people for extramarital sex is wrong.
Being a single mom is fine. Even if it shouldn't be encouraged, we should still celebrate those women who are struggling to make ends meet and vilify anyone who would critique or shame them.
Likewise, abortion is fine. In fact it should be encouraged. Shaming people for having abortions is wrong.
Getting divorced is fine. In fact we should encourage people in loveless marriages to get divorced rather than tough it out. Shaming people for getting a divorce or being divorced is wrong.
Being a 'slut' is fine. In fact we should encourage women to embrace the label and lifestyle. Slut-shaming is wrong.
Likewise, a woman being single and 'independent' into her 30's is fine, in fact we should encourage women to forgo marriage, children, and other familial goals in favor of their careers and social lives. Women shouldn't be judged for not being married by 25.
So at a minimum they seek to remove the stigma associated with certain behaviors that cut against family formation and they actually seem to encourage people to engage in these behaviors more often.
So lets say that someone takes all of this to heart, and thus feels absolutely no pressure to abstain from sex, get married before having kids, or even to get married at all, and even if they get married they feel they can get divorced without any guilt, and if they do happen to get pregnant then they can freely abort without a care...
On the margins, we would expect this to result in fewer people forming families, as now social norms forbid shaming or critiquing people for being unmarried, promiscuous, or otherwise avoiding the standard path towards family formation. Its a simple shift in incentives.
And even though the liberal 'elites' are still sticking to the normal path of family formation, they continue to push a social order which removes the pressure to form families, suggesting that any sort of societal pressure towards forming a family is oppressive and wrong-headed.
And so it really looks like they don't really want people forming families, if only because they decry any attempts to encourage family formation via negative reinforcement. If you truly thought family formation was an ideal for most people, if you thought it was part of the bedrock foundation of civilization, you wouldn't be actively promoting 'alternative' lifestyles as just as good or better than families, and you probably wouldn't attempt to remove all the stigma around alternative lifestyles.
Of course, that doesn't mean that shaming is the ideal method of promoting family formation.
And social conservatives have used social pressure to enforce many norms that, in retrospect, were indeed quite wrong-headed. Stigmatizing homosexuality and interracial marriage, for instance.
But if social conservatives aren't allowed to say "no, starting a family is actually the best route for most people and we should be willing to celebrate those who follow this path and also to stigmatize those who actively avoid this path in hopes of pushing them back onto it, for both their own good and the good of society," then yeah, they can suspect that you aren't actually pro-family.