How do we know that Christianity has NO PROBLEM with slavery? Just read the Bible:
Ephesians 6:5-9: Slaves, obey your earthly masters with respect and trembling, in singleness of heart, as you obey Christ, not with a slavery performed merely for looks, to please people, but as slaves of Christ, doing the will of God from the soul. Render service with enthusiasm, as for the Lord and not for humans, 8 knowing that whatever good we do, we will receive the same again from the Lord, whether we are enslaved or free.
Colossians 3:22 Slaves, obey your earthly masters in everything
Titus 2:9 Urge slaves to be submissive to their masters in everything
1 Peter 2:18 Slaves, be subject to your masters with all respect