And it's not just the Confederates. I've been hearing that chant from the right pretty much my entire adult life.

That's probably a function of when your adult life took place.

It is a gross oversimplification, but not an entirely wrong one, to say that from WWII until a short time ago, the federal courts were mostly applying the Constitution to tell state governments that the Constitution prohibited them from doing conservative things, and instead had to do liberal things. From the 1940's through the 1960's, and especially during the Warren Court, the SCOTUS started "incorporating" the Bill of Rights against the States and issuing a large number of rulings that circumscribed State restrictions on a range of issues. The Constitution was interpreted to ban segregation, prayer in schools, bans on birth control and marital aids, limits on free expression, and bans on abortion. That's also the period where the federal government also extended a lot of it's regulatory ambit, again mostly in service of things that modern liberals wanted and conservatives didn't - environmental regulation with the Clean Water and Clean Air Acts, the Voting Rights Act, etc.

It's no surprise that views of the proper role of the federal government ended up aligning the way they did.

Now both sides are a little wrongfooted on that question. For the first time since the 1930's, you're seeing some of the major changes in federal jurisprudence going the other way - the Court telling states that they lack the power to do things liberals want, like regulating gun control (or striking Trump from the ballot). As that trend continues, we are probably going to see some scrambling of positions on federalism....