It would be simplistic to reduce everything to slavery or race, but there have been times when such issues weren't on the political agenda, when Southerners could be quite radical. The tidewater "Cavalier" or planter interest has often tended to conservatism, particularly when change threatened it, but it's not clear how much such conservatism would have had with contemporary American conservativism if the planters had really had their way. It might have meant a much more elitist society than most of us could tolerate.
Southern unity in politics presupposes an external opponent. Otherwise, internal divisions would prevail. Upcountry "Celts" were quite willing to oppose the lowland planters when controversies with outsiders didn't prevail over local conflicts.
The idea that it was an underlying theological bent that made the decendants of the Puritans more amenable to liberal secularism than those of Anglican cavaliers is an absurd one, as a look at British history shows. As is the idea that Northern and Southern Baptists and Methodists split for theological or ethnic, rather than political reasons. It was the fact that Southerners felt the need to circle the wagons against outside influences made them more conservative than Northerners, who gave up such defensiveness over time.
Similarly, there may be a Celtic unruliness behind some Southern attitudes but a look at what became of Celtic Scots and Irish and Welsh at home in the British Isles, and in the Northern States, Canada, and Australia confirms that there was no Celtic drive towards conservatism. Such unruliness takes different forms in different circumstances, and is as likely to be radical as conservative.
If the country really is deeply divided culturally, it's a dangerous portent for the future. It's a sign of a healthy society that all the divisions don't run along the same fault-line, that some states in each region vote for a different party from their neighbors. When two regions like up against each other as solid blocs, as the 1860 election showed, things can get dicey for the country. It's probably the Civil War that brings out this feeling, though. The traditional Republican lock on the Mountain and Prairie states or Democratic strongholds in New York, Southern New England and the rustbelt don't cause such worries, because they never led to war.
In recent years, Liberalism or progressivism have become too closely identified with the two coasts and the arrogance of urban cosmopolitans. The rest of the country came to look on liberals or leftists as alien or hostile. It's possible that if conservatism is too closely identified with the South, it will lose support in other parts of the country. According to some observers this has already been happening.
The more an ideology tends to become associated with some specific part of society, the more likely it is to grate on others. There is a spiritual pride that grows out of strong political conviction -- of whatever stripe -- that can have a bitter taste for outsiders, when it's too closely associated with a specific class or race or region or ethnicity.
You could make a good case that the Middle West is the true heartland of American culture, but if a Middle Western chauvinist school were to be established, if people were to think of themselves more as Middle Westerners than as Americans, it would be as likely to offend and turn off other Americans as Northeastern, West Coast or Southern chauvinists are now.