Favors economic policies that make kids unable to support their parents into old age, furthering the decline of the extended family. Favors the idea that each American should be economically independent, an idea almost soley responsible for the smaller and less cohesive families we see in America compared to the rest of the country
Obviously you meant the rest of the world there, but that's not an issue.
But, sorry, nuclear families have nothing to do with USA / Republican policies and everything to do with industrialization. Nations which industrialized early have smaller families and less of an extended family thing going on than nations which industrialized recently.
For example, there's not a whole stack of extended-family type culture going on in Germany or the UK either (it's not that different to the USA), despite those places having significantly better welfare policies. Having good unemployment benefits also leads to a culture where people don't feel they need to rely on extended family, so I'd say neither side of the fence of economic policies breaks up extended families. In fact a "no government assistance" approach probably increases the amount people rely on family, rather than decreasing it.
The difference is that places like Greece, Spain were largely agrarian until even after WWII, so within living memory. The Italians who went to America might have more of an extended family tradition due to the fact that the vast bulk of the emigrants came from the rural south, not the more industrialized areas of Italy.
I mean pretty much the entire world except us and Europe. For example India is an industrial nation but still has the "families should stick together and cover each other's expenses" ethic. The adulthood cliff in America where you hit 18-22 and suddenly you're expected to be living on your own and supporting yourself doesn't exist in most of the world. The general standard is that grandparents help take care of their grandkids. And also that parents take care of their kids until said kids get older and then they're expected to support their parents. That's why you hear about immigrants always sending small amounts of money home to their families, because they're expected to help support everyone. Its also the standard in most of the world that an extended family will live together or, more likely, be consolidated into a 1 or a couple dwellings living in much closer quarters than people in the US would be used to.
Its a system that makes much more economic sense but also gives family members a lot of power over each other and would probably feel oppressive to most Americans. But you can see how now that most Americans don't have money we're being dragged kicking and screaming into a more internationally typical family structure even tho we don't want it. If I have any complaint about the US system its that in some cases it will give parents authority over their kids without responsibility for their kids. This is becoming much more evident now that the generational income gap has become so wide, as almost all younger Americans require their parents' support but there's no social pressure making said support mandatory. It provides the worst of both worlds: American parents still end up with a huge amount of power over their kids, but everyone is still trying to live as if kids - parents - grandparents will be 3 separate houses which makes no economic sense. Its ultimately yet another problem caused by the cult of continuous growth, of course your kids rely on you NOW but if you just wait they'll inevitably make more money in the future. Or on the flip side, of course you'll get that retirement money together sometime, there's no need to be a "burden" to your kids. And its the US destiny to keep growing GDP every year, there's no way we could possibly admit the American on the street has gotten poorer since the 50s...
And as for it not being republicans fault, agreed, but. When they say family values they mean anti-LGBT values, because the actual decline of the American family has nothing to do with what Republicans call family values.