The problem with that is drawing the line between what to throw away, and what to keep. For instance, I have a family member who is very liberal on a number of issues - too liberal for the United Methodists to allow for an ordination. Yet, she strict on other issues, on which she will boldly declare a Christian stance. I find her level of picking and choosing bizarre.
Let me try with Paul. What I depend upon Paul for is our earliest information on Jesus (e.g. the Words of Institution, and the Resurrection accounts), and how Jesus teaching, death, and resurrection affected the first Christians. He was the first person to think through the connection between Judaism and Christianity, with results that are quite basic to our understanding of what it means to be a Christian.
Where he’s simply reflecting his own culture, with no particular connection to Jesus, it’s of less interest. To give a non-controversial example, he speaks of incest. I have no reason to suggest accepting incest. But he just takes it for granted. Even pagans agree, he tells us. Similarly the sin list in 1 Cor 6:9 is pretty typical of Jewish ideas. He doesn’t explain the list or connect it with Jesus’ teaching. He just takes it for granted. His acceptance of slavery is another example. He had no particular reason to rethink that, though he wanted owners to treat their slaves well.
The hair length thing is almost amusing. He says “doesn’t nature teach”, and then ends up, well if anyone disagrees, that’s not how we do it. Most people today recognize that how people see things like covering your head and hair length is to large extent cultural.
When he gives serious attention to rethinking things, he can be quite radical. Eating meat sacrificed to idols is a well-known example. That’s an area where he departed from his Jewish background. And other Christians didn’t necessarily follow him, as we see from Rev 2:14,20.
Jews have had to deal with this issue as well. Even in the 1st Cent the situation was at times different than in OT times. Thus Mat 16 and 18 refer to the power of loosing and binding, which is the authority of a rabbi to make interpretations. I certainly don’t think the Church is inerrant. But the Christian community has the responsibility to help us decide what’s appropriate for today. I have reasonable confidence in the judgements of the mainline community.
You may ask, why the mainline community, and not either Catholics or conservative Protestants? Because the other major communities don’t really use that authority. They try to avoid using the power of the keys, and claim either unchanging tradition or literal interpretation of Scripture (which turns out to look a lot like unchanging tradition)
. That means that they don’t do the same kind of ethical investigations that the mainline community does. (Actually, Catholic ethicists do, but the hierarchy tends to ignore them where they conflict with tradition.)
This is mostly about ethics, because that tends to be where the hot issues are. But there's also a different approach to theology. I don't have time now to do a review of that, and in many respects CF isn't a place it can be done anyway.
