I think the Bible is a lot like that. There's no mistaking that American culture draws a lot of itself from the scriptures. Think about some of the idioms that originate in the good book:
A man after his own heart
Fight the good fight
Give up the ghost
Put your house in order
Writing on the wall
One way the Bible has sway is through a 'literal' understanding. (Ignoring the metaphorical and poetical parts for being impossible to take literally.) But that understanding is divorced from it's historical and cultural context so that it can be married to our lenses that tell us exactly what we will see and understand.
That way we won't be confused when Jesus says something that is the total opposite of what our interpretation seems to say that he said. There's the script of what he actually said, the script of what we think he actually meant; they're not always the same. Jesus' teachings were based on being compassionate, helping people in need, reaching out to outcasts, giving to the poor, visiting the ill, those in prison, and putting others ahead of ourselves. Any gospel that doesn't have that sort of sway to make us better people is not the gospel of Jesus.
The not the gospel of Jesus is a powerful tool that can be used to sway others - influence them, control them, make them afraid, and hurt them. The teaching about authority and submission is such a thing. It goes against the grain of Jesus' message. Jesus said to serve others, not to give out orders. He said to be last, not to be first. The confusion of lenses creates a situation where people will mishear and misapply even the most perfect sermon because we're predisposed to do such things as power corrupts, absolute power corrupts absolutely, and spiritual power corrupts spiritually.
I chose 'to hold sway' to remind me that my words are a powerful force that can influence people for good and for ill. For some, my words will sound like the latter, as if evil itself is trying to dominate the conversation and spread poisonous ideas. For others, my words will sound like the former, like a cool, fresh spring in the burning desert, to bring healing and restoration. With some groups of Christianity, it's one or the other, black or white type thinking with no room at all to consider if grey exists at all.
Here on out though, I can be a force for change. So bear with me, or not. It's up to you. I can't sway anyone who doesn't want to be swayed. Some people are so firmly entrenched in an opposing attitude, that there's no way they can be moved. Some people have probably seen something that begins them down the path of asking questions. Others are seekers who want all sides of the issues presented before choosing one. I hope I can be of help.
The toughest thing is understanding that everyone has a bias when it comes Scripture, how they were taught it, how they understand it, how they share it. A famous example is the Parable of the Prodigal Son, Americans almost universally leave out the detail of the severe famine in the land. To us, we see that the ungrateful, wasteful jerk squandered every dime of his inheritance on vices such as alcohol and women. We have no idea what a famine is or how hunger feels.
Learning what our biases are and how to leave them behind is a difficult process. Being aware helps, but soon you get curious. If Jesus was crucified about the year 35 A.D., then what other historical events were going on at the same time? Does the culture of Jewish society or Roman society have any impact on the story whatsoever? Does not knowing the culture and historical context provide for a superior interpretation of the events in the passage being studied? Asking the right questions is a good start, but seeking out the answer no matter where they take you is the next step. How does the Bible sway who you are and what you believe?