It would not have taken an amendment if not for Dred Scott. It could have been done simply by Congressional resolution. And again, the law books mention slaves, not freemen. Where, if not politics, did Taney justify denying citizenship to free blacks? It surely was not from the Constitution.
Oh wait, Dred Scott was determined to still be a slave by a 7-2 ruling, not a 5-4 ruling. The outcome of a new case might not have been a certainty, even with the Lincoln Justices and the Radical Republicans not letting President Johnson bring the Court up to full strength for fear of affecting decisions in a way they did not approve. (Which the Radicals did, by the way. What present day party does that remind you of?)
The Dred Scott ruling determined whether blacks, either slave or free, had been considered citizens at the time the Constitution had been written. It cited various state rulings and colonial laws and found that they had not.
Perhaps those arguments were persuasive enough to have made any subsequent court ruling that overturned Dred Scott simply a political decision rather than one founded on legal arguments and thus a bone of great contention. A constitutional amendment might set to rest legal challenges to the contrary, even if they had to hold guns to the heads of the Southern states for such an amendment to have been ratified. Well, it is still the law of the land, so the gun to the head approach worked.