In this post I'm just sketching the lay of the land.
i) The Duggar scandal has drawn attention to some related issues. There's an overlap between the homeschooling movement, the Quiverfull movement, and the Patriarchy movement. These are separable, and they are often separate in practice, but some members of one group belong to two or all three groups.
ii) I think much of the hostility towards the Duggars is due to the fact that some young liberals find the very idea of childbearing offensive. An antinatalist philosophy is taking hold in secular circles. Couples who prefer dogs to kids. They disapprove of childbearing. They're into "saving the planet" from "overpopulation."
iii) I don't know much about the Patriarchy movement. I'm guessing that Doug Wilson is probably its best representative. By that I mean, if you wish to evaluate the Patriarchy movement, he's probably the most articulate, responsible, and nuanced spokesman. I expect that he makes the best case that can be made for it.
I believe the Patriarchy movement intersects with the "Christian courtship" movement–as an alternative to dating. Likewise, the headcovering movement.
There's a "family" of positions.
I believe the Bayly brothers straddle some of these positions.
Keep in mind that the liberal media and "progressive Christians" use "patriarchy" as an abusive synonym for complementarianism. The go-to site for complementarianism is the Council on Biblical Manhood and Womanhood.
iv) I don't know much about the Quiverfull movement, but even though it's Protestant, it seems to take the Roman Catholic position to its logical conclusion.
Although I disagree with his position, I think Bryan C. Hodge has made what is the most scholarly case for traditional opposition to contraception, from an evangelical perspective.
v) Critics of complementarianism can point to cases of abuse. That, however, needs to be counterbalanced against horrors stories involving feminism. In a fallen world, whoever has power is in a position to abuse their authority. In a fallen world, abuse of power is inevitable.