Well, I would suspect that the US is much more culturally diverse than Russia. Something like a quarter of all US residents are foreign born or have a foreign born parent; that's reflected in cities like NYC. I am doubtful the numbers are similar for Russia, which is very large and does have some ethnic and cultural diversity, but has been dealing with the same groups for much longer and probably in more of an authoritarian way.
Not a defense of America; I do think the difference is more complicated than just the size.
[Insert default arguments of why America is special]
1.) big and sparse. oh wait..
2.) culturally diverse. oh wait.. [1][2][3][4]
I only have one explanation for social service scarcity in the US: It seems to be dominated by a culture where you'd rather have large parts of the population suffer rather than having anyone chip in for the bill of groups of people (s)he doesn't identify with. It's the idea of self responsibility pushed to an unhealthy extreme. At the same time when talking to people in the US personally (almost) everyone seems to be helpful and nice, much more so than in most European countries actually. That dichotomy is something I'm really struggling with, i.e. I fail to find a good explanation.
That's a very different story than the Russia link you provided, which, as I suspected, says that the country is mostly a) ethnically Russian and b) the rest are minorities the Russians have been dealing with for most of their history. I've even seen suggestions that the "White" category is perhaps too coarse, with observable cultural and political differences between whites of English descent and whites of German descent.
There's another pertinent datapoint, which is that Russia has a very low TFR and is experiencing a population decline and is pursuing policies to reverse that, as it is seen as a major problem. The US, on the other hand, continues to grow, largely because of immigration; policies to increase the number of children are superfluous.
That said, as I mentioned earlier, I think that's just a factor. Every country is special and has to be considered on its own terms. Is geographic size a factor? Probably. Is population size a factor? Probably. Is population diversity a factor? Probably. Is some remnant of the mythological self-reliance of American frontierism a factor? Probably. Is the rivalry with the USSR a factor? Probably.
Moreover, while I personally would like at a minimum a year of paid parental leave, the idea that parental leave represents Forward Progress and any country that doesn't have it is backward and deserves to be shamed is rather disturbing. Instead of trying to shame America to conform to somebody elses' norms, we should probably work instead on convincing them that it's a worthwhile and workable idea, and that the potential tradeoffs are worth making.
Not a defense of America; I do think the difference is more complicated than just the size.