You're viewing this from an American perspective. Slavery
did get abolished in Europe as a result of Christianity, from late antiquity into the early Middle Ages. The eradication of slavery can be pretty closely mapped to the spread of Christianity, which is why slavery within Europe was more or less history by the end of the first millennium.
Whether this had anything directly to do with Jesus or the Bible is another question, of course, but for whatever reason, Christians came to regard slavery as incompatible with Christianity, and they did this in pre-modern times.
I know less about modern slavery. However, it's worth pointing out that not only did key Enlightenment figures such as John Locke actually work in the slave trade, but slightly later key Enlightenment figures such as
Voltaire,
Hume, and
Kant developed extreme (for their day, not to mention for ours) racist views about non-Europeans, and about Africans in particular, which can hardly have been conducive to the abolition of slavery, and which helped to create an intellectual basis for the imperialism and colonialism of Africans by Europeans that happened in the following century. Modern "biological" racism really stemmed from these figures, who were extensively quoted by nineteenth-century racist anthropologists, and from the polygenesis theory, which was popular at the time among Enlightenment intellectuals partly because it contradicted the Bible. (That's one issue where it turned out that traditional Christianity was right and anti-Christian rationalism was not only wrong but morally very dubious.) So my point here is that one shouldn't make out that the Enlightenment sages were all on the side of right on these matters. On the contrary, the Enlightenment provided the intellectual justification for the age of colonialism.
On the original question of what people should make of those passages in the Bible that endorse slavery, not to mention other morally dubious bits, all I can say is that that's something people have to work out for themselves. Personally I'd say that the fact that the Bible says such things seems to be a pretty good reason for
not taking it as some kind of monolithic source of morality. However, that doesn't mean it's worthless. People forget that the Bible is not a single text but a collection of all sorts of different stuff. The fact that the author of Ephesians (who passes himself off as St Paul, but isn't) seems to endorse slavery might tell you all you need to know about the author of Ephesians (and perhaps of those who take Ephesians to be morally regulatory), but I don't see why that should colour your view of (say) the author of Matthew, or of Isaiah, or whatever.
It seems to me that pretty much all Christians regard some bits of the Bible as morally regulatory and other bits as "contextual", i.e. relics of their time not to be taken as binding today. Some of them are even capable of making quite artificial distinctions between Old Testament laws they like and others that they don't - e.g. I've encountered Christians who claim that there's a "moral Law", which is binding, and a "ritual Law", which isn't, despite the absence of any such distinction in the Bible itself (and indeed the book of Galatians, the argument of which is that you
can't cherry-pick the Law). The more honest liberal approach is to see the various books of the Bible simply as people's attempts to understand the divine and moral realms, with varying degrees of success. The author of Ephesians was wrong about slavery, as far as we can tell, but he might have got a few other things right, and perhaps other biblical authors did better on this matter, e.g. Isaiah 61:1.
