I'd say rather that both left and right have a complicated relationship with the federal system; I know plenty of lefties that are gratified, in the face of national pushback against their preferred initiatives, that they can at least make progress at the state level. And we have all seen evidence of even the most states-rights-rah-rah righties trying to make use of federal prohibitions and requirements when they thought that the national Zeitgeist was flowing in their direction.
I'm not calling anyone a hypocrite over this, by the way, and I think most anyone with a view on this would cop to a somewhat nuanced view if prodded.
Another is No Child Left Behind, a Republican-sponsored measure (though like DOMA it did receive bipartisan support) that forced a lot of specific educational policy on various states that weren't very happy about it, but acceded because the alternative was federal defunding.
Another is various efforts involving abortion and, lately, contraception; I hesitate to get too deep on specifics lest I derail the thread, but right-wing efforts there have not been confined to the state level. (Nor have left-wing efforts, obviously.)
It's not about the fact that municipalities can have different laws, which is not unique to the United States, and I'm curious if you have evidence that "the modern left dislikes" the fact that cities and states can have varying laws. I don't have that impression, but it does make for a convenient straw man.
The left and the right both support state and local experimentation when it suits their desired outcome and they both support national power when it suits their desired outcome. The political parties may pay lip service to political philosophy, but in practice they take advantage of whatever philosophical argument works best for the outcome they wish to achieve. See: slavery, fugitive slave laws, segregation, voting rights, same sex marriage, marijuana legalization, abortion, labor organization, union busting, immigration, environmental protection, consumer protection, universal health care...
Part of the glory and endurance of the United States is its dual republic[1]. It oxygenates conflicts and allows for experimentation. The modern left dislikes it and prefers federal power, but if I had to take a bet on whether the United States will exist in 500 years, it'd be premised in large part on the exquisitely balanced dual sovereignty of state and federal governments.
[1] http://thomas.loc.gov/home/histdox/fed_45.html