What you outline could happen, but there's no reason to think it will in the foreseeable future. Don't expect Millennials to be good little social democrats, either - they don't trust and don't like government or government management of the economy, as multiple Reason-Rupe polls have found; see
this excellent report for the real story on Millennial politics. Millennials may bask in social liberalism* but they also like business and leaving economic lives alone, and only generally favor programs that are fairly universal, don't cost them too much, and can't be used for graft or manipulating them; this excludes most of the current government, but still leaves a few new possibilities open (e.g. expanded college grants).
I'd venture that if anything along the OP's lines does happen, the labor component is much more likely than the social welfare component. Labor movements, often strong ones, have grown natively and spread from sea to sea many times in American history. Social welfare has never done that, but rather was a European creation that established itself in universities and elite Northeastern circles** and has never penetrated much further. Historically progress in social welfare has been
accepted but has never been
demanded like progress in labor was, and was accompanied by doubt over national vitality (1930s/40s***) or an attack of national guilt (1960s); outside of those times and the programs they spawned social welfare has been not core but very peripheral to the American system.
*At least in the old culture-war sense; they don't let that stop them from leading a life of (relatively) tight morals, even compared to previous generations. They're icy towards casual sex and don't enthusiastically accept homosexuality (
source), and
sleep around less than their parents did (8 partners at age 25, compared to Xers' 10, Boomers' 11 (talk about promiscuous hookup culture
), Silents' 5, and GIs' 3).
**Not coincidentally the parts of America most closely tied to western Europe. Marxism (famously) and Fascism (not so famously) traveled a similar pathway, from western Europe to American universities and some Northeastern elites, where they stayed until they died out. In Marxism-Leninism's case, it likely had more American adherents inside colleges than outside them.
***Even then modern social welfare forms didn't penetrate much until WWII; the 1930s government disfavored "the dole" and favored "work relief" (that's what the WPA and CCC were for). Social insurance is a different matter; Americans seem to like that, and even Founders such as Thomas Paine did early work on it. They still don't demand it spontaneously, but they often do just that when it's presented to them, and typically demand to keep it. That's more than can be said for social welfare.
So in short, Americans tend to be intermittently enthusiastic about labor issues, really like social insurance when it's presented to them, and don't like social welfare that much but will accept it at times. This is a somewhat different pattern from Europeans, but we should expect that. Europe's political development path was the result of peculiar historical circumstances including two devastating wars, a great depression, the iron curtain, and going leftward in the 1970s crises*. The widespread assumption that as countries develop they should become more like Europe was after all that is simply not logical.
*In particular, "every other country" having universal health care is a newer development than people think. As recently as the 1970s it was mostly confined to a scattering of Germanic and Commonwealth countries, and "every other country" having it only became a plausible talking point after 2000 or so.