None of the passages you mention forbids slavery.
The 1 Timothy passage forbids slave trading. Yes it's an important distinction---slave trading often involves related kidnapping and violence. This passage is silent concerning a slave who voluntarily became one to repay a deb.
The 1 Corinthians passage simply acknowledges that being a free man is preferable to being a slave. But in fact, it clearly states that if you can't gain your freedom, you should "not let it trouble you".
Finally, the Philemon passage simply states the superiority of the Christian brotherhood relationship over the slave/master relationship.
These passages are even less relevant, then, than the various passages in the New Testament which command slaves to obey their masters and masters to treat their slaves with respect.
Indeed, the one specific prohibition of slavery in the Old Testament is that an Israelite was not to enslave another Israelite. Thus your later point, about not enslaving others because anyone can be a member of the new Israel, is a good one.
If someone becomes a slave voluntarily, then it could be argued that it isn't slavery as most would define it.
In any case, I think it's abundantly clear that slavery as practiced by the Western world through the 19th century was an abomination to God. In addition to the fact that you had forcible kidnapping from Africa (punishable by death in the OT), breaking up of families by separating parents from children (clearly a sin in the eyes of God), you had Christians enslaving other Christians. I really run out of pejoratives to say how bad this was.
In the New Covenant, anyone can be a believer, and we never know when the grace of God will come upon the worst unbeliever, so slavery is always a sin no matter where it is practiced. So I think the point is really moot today.
But we also need to remember that the fundamental purpose of the NT was to show mankind salvation in Christ, and Christian living. Changing society comes after that, and is a natural outgrowth of the former.