Christ's teachings are indeed the ideal we live by, but the change takes place internally in how we relate to others.
And if enough Christians together are living by Christ's commandments, guided by the Holy Spirit, then you will start to see society as a whole reflect a more ideal and just world (Although it will never result in perfection, because we won't live in a perfect world until Christ returns); which is the point where a Christian society looks around and says "you know this slavery thing just doesn't feel right, I think we should get rid of it."
The misconception would be believing that the law of Moses, which did not condemn slavery, represented the ideal by which the world was to live by.
It was a law given to a specific people, for a specific period of time, to serve a specific purpose in the plan of salvation.
Slavery at the time was a key part of the justice system, not just the economic system.
If you had to pay restitution to someone, but had no means, then you become enslaved to them for a period to work off that debt.
The law of Moses was actually a means of making what was at the time a necessary institution of order be more humane and fair, by laying out standards of behavior in addition to setting timeframes in which slaves were to be freed.
If the American south had adhered to the biblical guidelines of slavery, then the institution never would have lasted long as each generation kept being freed (During the Jubliee, every 50 years, all slaves are freed and all debts are forgiven). Nor would they have been allowed to beat and abuse their slaves.
You then begin to see, in that context, how the law of moses was actually a system that represented an improvement over the world system at the time, being more fair and just than it otherwise would be.
[Having a single master who has dominion over you is much more intimate than being a part of a system where your labors are exploited for others' monetary gain.
It's really a matter of perspective, because you'll see some socialists/communists who try to describe capitalism in slavery terms (a system where a person is being exploited for monetary gain and supposedly has no say in the matter).
They will point to "sweat shops" of the past, or in other parts of the world, and complain that it is little more than a system of "modern day slavery'.
The point being: No world system is ever going to be viewed as 100% fair and just from a purist's perspective. Certainly no system we can devise here on earth will ever function according to God's perfect standards of justice and righteousness. Standards keep trying to evolve to a more ideal world, but it's never going to happen until Christ returns. Therefore, we can't expect that God's laws to Moses are intended to reflect an ideal system of law that the entire world is to adhere to for all time.
An example of the law being a compromise for the condition of man, and not a reflection of God's absolutely perfect will for us, would be in the fact that God originally intended for there to be no divorce, but recognizing the hardness of men's hearts allowed it to them as a necessity in some cases.
If god were to have given the Israelites a law that represented the ideal purity of eden on earth, then they would have been utterly incapable of adhering to it, and they would have been unable to thrive in the world system as it existed.
God gave them a set of standards that were relevant to their context.
And I suspect that if God were to speak to us today, he would give us similar guidance in the sense that he would take into account the realities of the world system we live in and tell us how we can make the best of it in a way that is more fair and just than otherwise would be expected.
Today he does do this through the guidance of the Holy Spirit, but it only begins to be reflected in society when a majority of the population is being equally guided.
That, I believe, is the secret to why the western world began to thrive ahead of everyone else after the protestant reformation, when the word of God became more known to the populace at large.