For man does not originate from woman, but woman from man... (1Corinthians 11:8)
God had a purpose in creating Adam first and His purpose is not complicated.
Creating Eve second established order between the sexes. Man is the glory of God and woman is the glory of man. Man is to be responsible and woman is to help man.
Sadly, we're so contaminated by our culture of rebellion that we turn into weasels...
We claim male authority is a concession to sinful men that Godly men will rise above. Thus Tim Keller says the husband's authority over his wife is only "tie-breaking" authority. So what kind of husband allows the argument to get to that point? The Godly man would have deferred to his wife long ago.
We claim male authority is only applicable in the home. Thus Roger Nicole taught women could be pastors and elders in the church while at home the husband remained the head of his wife. (Later in life, Dr. Nicole became more consistent, denying male authority in the home, also.)
We claim that adhering to male authority outside the home and Church weakens male authority in the home and Church. In a recent e-mail explaining why he was working to add women to his college's board of trustees, a friend explained: "(By maintaining male authority on our board of trustees) I think I weaken my arguments for restrictions where they are Biblically warranted, such as in the authority structure of the family and of the church."
We nitpick, claiming this applies to that over there in every case that arises on the third Sunday of the seventh Thursday following Good Friday, while the other thing has to do with cases in which the hypotenuse is squared by the orangutan as long as two-thirds of his body is under the perambulator. We diagram, parse, and divide domains, structures, and spheres until those nasty "restrictions" are all that's left.
We've eviscerated manhood and womanhood of the Creation Order, leaving sex a petty thing having to do only with certain restrictions pertaining to church pulpits, session meetings, and body parts (if you're called to marriage).
As I never tire of repeating, Christians of all previous ages would universally condemn our faithless denial of the application of God's Creation Order to all of life. Further, they would scorn our hypocritical attempts to pass this faithlessness off as a mark of our zeal to guard the few "restrictions" remaining in the New Testament.
Wherever man is man and woman is woman, God's Creation Order applies. It is no special revelation for God's Covenant People. It is the universal law of mankind.
We may disagree over how to witness to it or how and when to seek its implementation, but the fact of it is as indisputable as life-long fidelity and heterosexuality. To fail to testify to male authority is to be ashamed of our Lord and His Words.