First, I'd like to make it clear that I'm not talking about "the patriarchy" which oppresses modern women with deodorants and air-conditioning, but rather about so-called "patriarchy movement" which became popular among fundamentalist Christians in the USA, especially in home-schooling circles.
Are there problems with this movement? I'd say yes. It appears to me that their ideal society is not a New Testament Church and not even an Old Testament Temple but the Bronze Age nomadic existence and being totally withdrawn from any form of society, hence their accent on the role of the husband as the king, priest and prophet of the family. Their ideas about marital obedience are more extreme than those of Muslims:
Men are in charge of women, because Allah hath made the one of them to
excel the other, and because they spend of their property (for the
support of women). So good women are the obedient...
says the Quran.
As you can see, female obedience is conditional upon men financially providing. I heard that in some Islamic countries the failure of the husband to provide a living is grounds for divorce. Yet, the supporters of the patriarchy movement will often tell you that it doesn't matter if your husband is an unemployed alcoholic, you still have to obey him in literally everything, including the colour of your shoes.
They are also very suspicious towards the Church. The Scriptures teach us that we should not be forsaking the assembling of ourselves together,
as the Church was founded by Christ and is His Bride, and yes, I know many churches are corrupted, but by no means all of them, yet to listen to some of the people talk, there is no church to satisfy them (apparently they consider themselves to be in sinless perfection).
Of course, being a church member means being under the church authority and it "undermines" the authority of the husband in some ways, so the patriarchy adherents can't agree with it. For instance, if you are a Roman Catholic woman married to a husband who isn't a Catholic and he asks you to use anti-conception, you can't do it since your church officially teaches it to be a (mortal) sin and your church's authority will overrule that of your husband. The same holds true about education since a Catholic promises to raise his children in the Catholic faith.
I could give you more examples, but I guess you know what I mean. Another thing to consider is that the nuclear family doesn't exist in the vacuum, like many of these folks would like in their libertarian fantasies. There is extended family to take into account and the laws of the society. You can't just go live in a tent like Abraham did. That extends to the marriage rites. In every Western country there is an established form of getting married, acknowledged by both church and state, thus the ideas of some men that you could just live together "but be married in God's eyes" don't hold water.
There are more things one could say about them, but that's just a couple of points which first come to mind.