From my perspective, it tends to be the people who say they support "family values" that actually support legal and social measures that keep families small.
Look at who it is defending zoning laws enforcing "single-family household" status as excluding larger chosen (non-blood-related) families, and compare to who it is embracing legal and social norms that allow maximum flexibility in assembling a strong, self-supporting structure from such components as available. Look at who is trying to restrict legal marriage and adoption and who is trying to extend it. Look at the group voting for judges that view large aggregated families-of-choice as evidence of perversion -- from which children should be protected -- and the group voting for judges who view a large, stable support network built from people who love and care for each other as precisely that. I'm all for "family values", in by that one means values that support large and strong families... but if I say "family values" in public to a random stranger, what's going to come to their mind is not the same as what I'm actually referring to.
I say this as someone who is overwhelmingly happy to have participated in the upbringing of children -- two of whom are now legal adults -- in whose genes I have no role, but to whose memes and ethics I am gratified to have contributed. I'm glad to have contributed to the financial stability of their household; I'm glad to have been another person there to help with homework and listen to their stories and serve as a role model and help keep things running. The people who say they support "family values" but who would have broken apart that family? I cannot, at such short notice, find words for the damage I see being done -- or attempted -- in the name of "family values".
And yes, I know that you're acknowledging much of the above, and that a great deal of my rant (perhaps all of it) doesn't apply to you. Please forgive that. I don't believe your assertion that anyone (for a statistically significant value of same) views state programs as an adequate replacement for having a genuine support structure... but would suggest that, perhaps, there are those who would like those who don't have a support structure to have somewhere to turn.
I've known too many people whose blood families weren't a healthy place for them -- physical abuse and the like. Several of those people were welcomed into a family of choice that gave them the support that they needed -- but not everyone can be that lucky, and establishing social policy in a way that only helps those who are already fortunate... well, there's a lot of that done already, and a lot of people it leaves behind.