Now we discussed how Jesus went against Jewish and Roman culture. He taught against divorce, even though the law made exception for it, due to hardness of heart, and even though some schools of Jewish thought allowed it with little regard for the woman. Jesus had no problem calling out wrong culture, and teaching what was true, and He did so from Eden, as you note.
Likewise, Jesus had no problem with having female disciples, who learned at His feet, which would be culturally scandalous. He still did it, and did not worry about culture.
And Paul had no issue stating that we should not be slaves of men, that you should buy your freedom if you can, and appealing for a slave to be freed, because he saw slavery as wrong, even though the culture did not.
If something was wrong they did not uphold it. But that is not what we see in the case of wives submitting to husbands. Paul and Peter both uphold it. Paul upholds it in a number of contexts. That is why I said you cannot say they are simply referring to culture if they make theological arguments supporting headship. The comments exist in a culture. But they do not go against the aspect of headship. They go against the Roman practice of tyrannical practices, by showing the example of Jesus.
Jesus is the head of the church. The husband is the head of the wife. The husband should imitate Jesus in the practice of headship, serving, and laying down his life.
This is a good chance to pick back up on the line of discussion we were engaged in earlier. You indicated:
I think this is where we are using terms differently, a bit.
I do not say that mutual submission is impossible in the parent child role, even though a child is not mentally able to lead. Mutual submission is an attitude, a mindset, of serving others, looking to other's interestes, etc. In this case all agree, due to the nature of a child, that it will be unequal submission. The parent has the accountability and leadership role.
But the there is mutual submission because the parent is looking to the child's best interests, putting aside their own interests for the sake of the child, etc.
In the even more extreme example, Jesus, though Head, and God of the universe, showed mutual submission to us, though He never ceased to be who He is. He showed how a loving Head acts, which does involve mutual submission, even within different roles.
That is why I was not arguing for one-sided submission. I have noted that there is mutual submission. And there is also headship.
So yes, it undermines "one-sided" submission. It does not undermine submission of wives to headship. But it does rule out the husband just acting on his own whims, by his own authority, without regard for the spouse, etc. His whole mindset is to be like that of the Head of the Church, Christ, in that He is giving himself up for her.
That does not undermine headship. Because the headship in question is patterned off of the Headship of Christ, who did serve, who did love ,and did honor. But He was still head.
Paul does not say that husbands are not the head. He says they should follow the example of the Head of the church, Jesus, in self-sacrificing love.
Well, we agree on part of that. They certainly did all they could to uphold the goodness of marriage.
What Paul's statements undermine is not headship, but the ways that headship is exercised.