The first woman was made as a helper. (Genesis 2:18,20)
Many in the church stretch this from Eve being a made as a helper, to all women being made as helpers.
This word helper is ezer in Hebrew. And ezer, most times in the Bible, refer to God as helper. The helper - God - is the authority figure acting as benefactor, doing what the helpee cannot do, not the assistant obeying orders. If helper connected to authority at all, it would have made women the authority figure. (The other uses of the word ezer are in Exo_18:4; Deu_33:7; Deu_33:26; Deu_33:29; Psa_20:2; Psa_33:20; Psa_70:5; Psa_89:19; Psa_115:9; Psa_115:10; Psa_115:11; Psa_121:1; Psa_121:2; Psa_124:8; Psa_146:5; Isa_30:5; Eze_12:14; Dan_11:34; Hos_13:9)
Your husband … will rule over you. (Gen 3:16)
Patriarchists believe "he will rule over you" mean "I, the Lord, want him to rule over you. I called him to rule over you." But if that is so, they should translate the other prediction of how women and their seed will be treated (enmity between snake - Satan - and the woman, and between thy seed and her seed; the woman’s seed hurting Satan and vice versa- Gen 3:15) the same way. In that case, God told Satan: "I, the Lord, want Satan to harm humans. I called him to harm humans." If men, on the Day of Judgment, will be praised for ruling well, then Satan will have to be praised on that day for following the prediction that he will harm humans.
Genesis 3:15-16 are, by their very wording style, not commands. They are predictions. Patriarchy is one more piece of evidence that the prediction in verse 16 is true.
Up to here, I can say without a shadow of a doubt that a patriarchal interpretation of these verses would be dead wrong. It is just not in there. If God wanted patriarchy, he did not say it there. Those in the rest of the discussion are less clear.
The head of the woman is man/ the husband is the head of the wife (1 Cor 11:3; Eph 5:23)
"Head" here comes from the Greek word kephale. Kephale has the first, literal meaning of the body part above the neck. After that, it spans a few figurative meanings. Not all Greek scholars agree that "authority figure" is even among the figurative meanings of kephale. At leaxt one authorative Greek lexicon does not mention leader or authority figure, or any term like that, as possible symbolic meaning of kephale. Among the other possible meanings are origin, source, prominent part, capstone of a pyramid, and provider.
Even those who want to see authority in these verses, like Wayne Grudem, see little of it. Grudem say that kephale denotes authority about 2,1% of the times it is used in Greek - and top, starting point (origin) 3% of the times.
Also significant is that these 2 texts say man is head already, and does not tell him to be the head. This could be a statement admitting the position men held in society at the time this was written, or it could mean that being the head (Origin? Source? Prominent part? Authority figure? Provider?) is a simple fact in some regard for even the most hen-pecked man, without him having to lead to be the head. It is hard if we cannot say for sure what head mean here.
Wifes should submit to husbands. (Eph 5:22,24; Col 3:18; 1 Pet 3)
There is no verse in the whole Bible where God say husbands should lead wives, or tell them how to lead their wives. Going from: "Wives should submit" to "husbands should lead" is like going from "You should turn the other cheek if slapped" to "I should slap you."
Almost all submission verses were (this is less clear in English than in the original Greek) written in a tense that is either middle or passive. It could be "wives are subject to husbands." It is not in the active form - women are not commanded to submit in them.
The Greek word submission has a meaning range from what soldiers did in the military, to letters being attached to each other in the postal system. As such, it could also mean they were attached to husbands, or that they were told to stay with the husbands.
And almost all of them are close to submission verses that include the other party. Ephes 5:21 say “submit to one another" before the verses just afterwards call wives to submit. Peter 3 calls wives to submit, and husbands to do likewise.
(Nothing in the Bible tell men to rule their wives. Nothing in the Bible tell men to rule their households, except:
Esther 1:22 He send dispatches to every part of his kingdom, … proclaiming that every man should be ruler over his own household.
But it is the heathen king Xerxes who gives this command, not God.)
Evidence that God want wives below husbands in a hierarchy is rare in the Bible. Men usually led in the worldly and sinful cultures of Bible times, but there could be doubt on the evidence that God wants this to continue.