I'd like to comment on a subset of complementarians who ground their position in the eternal subordination of the Son, which, in turn, is grounded in eternal generation. For discussion purposes, let's stipulate eternal generation.
i) The direct way to underwrite complementarianism is to say that while men and women are alike in many ways, and can do the same things in areas where they are alike, men and women are naturally dissimilar in certain significant ways, and social structures ought to reflect and respect those differences. Men and women have certain physical and psychological differences which, at least in part, undergird complementarianism.
ii) The question, then, is whether these natural differences are sufficient or insufficient to justify complementarianism. If sufficient, then the eternal subordination of the Son is superfluous to complementarianism. The natural differences between men and women are adequate to warrant different treatment. Treat like things alike, and unlike things unalike. That's a stand-alone justification for the position. It requires nothing else.
iii) But suppose the natural differences are deemed to be insufficient. In that event, appeal to the eternal subordination of the Son functions as a makeweight. If, however, the natural differences are insufficient to justify complementarianism, then it's hard to see how invoking the eternal subordination of the Son will shore up that deficiency.
On this view, what accounts for the eternal subordination of the Son lies in the intrinsic difference between Father's mode of subsistence and the Son's. The Father is ingenerate while the Son is generated. The Father is inordinate while the Father originates the Son.
But even if that's true, how does that contribute to the argument for complementarianism? How is the relationship between men and women analogous to eternal generation? Take this classic formulation:
The role of a father is “to beget,” just as the meaning of sonship is “to be begotten.” The Father, therefore, is unbegotten, but is origin and progenitor of the Son, who himself does not beget, for there is no “Son” in the Godhead other than himself. That is to say, the whole reality of the Father is to beget, to generate, to give all that he has, namely, his whole divine nature, to the Son. And the whole reality of the Son is to be begotten, to be generated, to receive all that he has, namely, his whole divine nature, from the Father...The life of the Father is an eternal giving of himself whole and entire to the Son. The life of the Son is an eternal receiving of the Father whole and entire.
That strikes me as a lucid definition or exposition of eternal generation. But surely the relationship between men and women isn't comparable to that.
Even if the subordination of the Son is intrinsic to the Godhead, that would be extrinsic to men and women unless it has a parallel in human nature. How can something extrinsic to men and women undergird complementarianism? How can something about the Father and the Son help to underwrite complementarianism unless it has a parallel in human nature?
iv) But suppose, for the sake of argument, that it does have a parallel in human nature. If so, then that's what undergirds complementarianism–not something about the Father and the Son, but the analogue in human nature. Yet in that event, the eternal subordination of the Son is superfluous to the case for complementarianism. That principle does no work. It's something about the essential constitution of men and women that's the differential factor, and not something about the essential constitution of the Godhead.