Whenever Paul is writing, we have to remember that one of his key goals is to instruct the Christians how to live in ways that do not hinder the gospel message by being disgraceful:
On the contrary, we put up with anything rather than hinder the gospel of Christ. - 1 Cor. 9:12
"... so that no one will malign the word of God ... and soundness of speech that cannot be condemned, so that those who oppose you may be ashamed because they have nothing bad to say about us. ... so that in every way they will make the teaching about God our Savior attractive. - Titus 2:5, 8, 10
In Phillipians 1, Paul explains why his humilitaing situation actually serves to advance the gospel. His primary concern is not with his own freedom, his own comfort, his own status or reputation; but with the gospel and it's reputation.
One school of thought considers slavery to be a cultural accomodation. God knew from day one that for the better part of human history, human nature would condone slavery. Aristotle believed that it was the nature of some men to be 'living tools', whose only purpose was to serve their master so that their master could take care of their basic needs. Some societies saw manual labor as beneath them, so they depended upon slavery to function. Some estimates of the ancient world was that 60% of the population were slaves. Did Paul instruct slaves to submit to their masters because he believed that submission was the foundation of Christianity, and the enduring princple to which all believers must obey until the end of time?
Let's imagine an alternate history of Christianity where Jesus' message inspired all converted masters to free their slaves: It was widely believed that people were slaves were the sort of people who just couldn't take care of themselves (that's why they were slaves.) Christiantiy would have been seen as a backwards cult that was doomed to fail because it didn't take care of people who should have been taken care of. At least, to the rest of the world, as slaves, their masters would have an obligation to give them shelter, clothing, and food. As free men and women, they'd be reduced to being beggars. This version of Christianity wouldn't have lasted very long. It would only be remembered as that one irresponsible cult with some funny beliefs. Not only that, but people would have rightly criticized it and that would have hindered it being accepted in a big way.
Okay, so Paul couldn't do away with slavery in his day and age - but what if the advice that he wrote about slavery undermined the institution from the inside out? What if the plan was that for one day, the institution of slavery would be irrelevant to Christian experience? Then ought slavery to be maintained so that the Bible's advice could continue? Or would it be better to let slavery go altogether so that the world can move ahead? In 1948, UN Article 4 of the Universal Declaration of Human Rights banned slavery globally. So if anybody wanted to restore slavery, they'd be breaking laws to do it. And that would bring critcism upon the gospel.
One school of thought considers the subordinate role of women to be a cultural accomodation. God knew that for the better part of human history, human nature would cause men to rule over women. Almost all over the world, women were viewed in a less than favorable light. They were never the equals of men. Aristotle believed that while a wife and her husband were both free, her rule lacked authority, and so it was the duty of the husband to represent the family in all public matters. Paul had to speak to the role of women in Christianity from the point of view of multiple distinct cultures. One culture might allow for more freedom, but to another culture that would be shameful. So Paul did ask women to submit to their husbands so that their actions brought respect to the individual households and the approval of outsiders.
Let's imagine an alternate history of Christianity where Jesus' message inspired all husbands to treat their wives with love and respect, so much so that leadership roles were opened up for women in the churches. I'd imagine Christianity is being viewed as yet another women's cult. Priestesses were not uncommon in ancient Rome. But just as much as there were cults for women, they often struggled to gain traction with the men who often preferred warrior cults. Christianity would be amazingly popular, but without influence among the leadership. In the end, it would fade into the mists of history as one of the women's cults that couldn't stand the test of time.
Okay, so Paul couldn't do away with sexism in his day and age - but what if the advice he gave for families was to undermine sexist attitudes from inside out? What if the plan was that one day, men and women would be so equal that nobody would think a thing of it women were pastors and men didn't have to shoulder the burden of leadership alone? In that world, wouldn't the teaching that only men must lead be a stumbling block to the rest of the world who would critize that attitude for being sexist?
So here we are, having rightly divided the word of God that slavery is not what God wanted until the end of time; but believing that the sexist attitudes are what God wants until the end of time. We are doing what Paul was afraid of, bringing disgrace upon the gospel by trying to uphold it literally. We are bringing shame upon it's teachings by denying some of them to uphold others. We're the reason why the gospel has the bad reputation that it does. Paul never suggested that the believers up and freed their slaves, but he hoped that one day they would. Paul never suggested that women be handed the keys to leadership, but he hoped that one day they would. At least we've got his lesson half-right. Hopefully we'll ace it in the near future.