Patriarchy places a very high priority on the family: Children obeying fathers. Wives obeying husbands. Schooling children at home. Staying with the family until marriage. Your father finding you someone to marry. Adherents are even opposed to Sunday school as the family should be teaching children about God.
What does Jesus teach us about family? Jesus is not nearly as pro-family as patriarchy supporters. He says, among others:
Mat_10:37 He that loves father or mother more than me is not worthy of me: and he that loves son or daughter more than me is not worthy of me.
Mat_19:29 And every one that hath forsaken houses, or brethren, or sisters, or father, or mother, or wife, or children, or lands, for my name's sake, shall receive an hundredfold, and shall inherit everlasting life.
Luk_14:26 If any come to me, and hate not his father, and mother, and wife, and children, and brethren, and sisters, yea, and his own life also, he cannot be my disciple.
Jesus tells his followers to call nobody father, as there is only one father and master over them:
Mat. 23:8-9 But you are not to be called “Rabbi,” for you have only one Master and you are all brothers. And do not call anyone on earth “father,” for you have one Father, and He is in Heaven.
As for honoring his family, which family does Jesus choose sides for? :
Mat 12:46-50 While he yet talked to the people, behold, his mother and his brethren stood without, desiring to speak with him. Then one said unto him, Behold, thy mother and thy brethren stand without, desiring to speak with thee. But he answered and said unto him that told him: Who is my mother? And who are my brethren? And he stretched forth his hand toward his disciples, and said: Behold my mother and my brethren! For whosoever shall do the will of my Father which is in heaven, the same is my brother, and sister, and mother.
A Christian, according to New Testament writers, has a new family – the family/ household of God. Among numerous such verses, here are a few:
Eph. 2:19 Now therefore ye are no more strangers and foreigners, but fellowcitizens with the saints, and of the household of God; Gal_6:10 As we have therefore opportunity, let us do good unto all men, especially unto them who are of the household of faith.
1Pe 3:8 Finally, be ye all of one mind, having compassion one of another, love as brethren, be pitiful, be courteous:
Believers are called brothers and sisters (meaning of brethren) – with one Father.
Now, if the New Testament extensively discuss being part of a spiritual family, why would God give authority to teach you and help you on the right way only, or mainly, to the natural family? Christ even approves of abandoning your natural family to follow him.
Biblical Patriarchy is not the way families are pictured in the Old Testament either – in the Old Testament, most men were not patriarchs. Patriarchs were tribal chieftains, and most men and their families had to submit to whichever son the old patriarch appointed as new patriarch. See, for example Gen 27:37:
And Isaac answered and said unto Esau, Behold, I have made him thy lord, and all his brethren have I given to him for servants; and with corn and wine have I sustained him: and what shall I do now unto thee, my son?