To answer your questions, I will clarify a couple of points. Slavery existed everywhere, not just in the west. It has been, historically, very ingrained into human society.
Now, it did not come to an end all at once (it even exists today). It’s decline has been a slow and laborous process. And like many things it has ebbed and flowed (2 steps forward, 1 step backward). But from the very beginning, we see Christianity being at the forefront of the decline. Starting with St Paul’s exhortations on the relationships between master and slave.
But when the big steps were taken, I do not just mean it was led by people of Christians cultures. What I am saying is that time and time again, it was led by the devout Christians of the western societies; those who took Christ’s two commandments very seriously. And the Christian countries responded. Not in perfect lock-step, not without a great deal of strife at times, but they did.
Yes slavery existed in the west, that was largely one of the “steps backwards”. Your questions can be summed up as “why was slavery practiced in Christian countries?”. Again, the question is simple to answer, it has always been practiced. It is a result of our fallen nature.
The important question is “why did it end?”. That tells you a lot more. It was not “after all this time”, it started at the very beginning of Christianity and you can find progress throughout the last 2000 years.