That is a stain that they won't be able to remove easily.
Lets point out the stains of the Democrats for once. Ones they will never be rid of. Secession, segregation, secularism, and socialism. This is a Democrat stain not a Republican one.
Oh come on, you're not even trying. Lets go down the list.
Secession: The Democrats of today's America don't agree with secession, and I would wager that they want the Confederate flag to go down more than Republicans at this point. The Dixiecrats are dead.
Segregation: Same as with secession; the modern Democrats support voting rights, a generally more multicultural society, and probably would support integrated busing if it were an issue now. Additionally, their candidates for public office tend to be more diverse (and therefore less segregated) than the Republicans. Also, it was a Democratic president that pushed for and signed the Civil Rights Act. The next Republican president was wary of busing and so forth.
Secularism: Ignoring the fact that secularism isn't that bad in many cases, the Democrats have pastors and religious people in their party. Heck, the state senator that died in the Charleston shooting was a pastor. And all modern Democratic presidents tend to say things like "God bless America" and so forth. The Democratic Party is more open and inviting to secular people and atheists though, so you get half a point for that.
Socialism: If you call wanting a working social safety net socialism, then sure. Though I will remind you that it was a Democratic president, Woodrow Wilson, that imprisoned thousands of Socialists before and during WW1. And it was generally Democratic presidents that waged war against Socialist nations during the Cold War (Korea and Vietnam).
As for the idea that the two parties don't have cultural issues anymore, there's still abortion. That'll be a hot-button issue forever. And the death penalty. And the drug war. And as of now, gay marriage is still an issue, if one settled by the courts.