It's a trend in contemporary theology to (a) claim that the doctrine of the Trinity supports complementarianism, (b) claim that the doctrine of the Trinity supports egalitarianism, (c) claim that the doctrine of the Trinity can't be used for either of these claims.
I've lately been reading Kevin Giles, The Eternal Generation of the Son, which for the most part is a solid defence of the doctrine of EGS. He's quite right, in my opinion, to tackle head-on evangelical scholars who reject this doctrine. The quality of evangelical scholarship in understanding this doctrine is half the problem. For example, Giles references Grudem, and Grudem treats the topic in an appendix (6) to his Systematic Theology, in which Grudem stunningly shows a degree of historical and doctrinal ignorance that is appalling.
However, Giles has a second agenda, which sits less comfortably. He regularly points out that the Father and Son are co-equal in essence and in authority. Now, this is quite true, but the second half of that statement hinges upon the Father and Son sharing all things, so this includes them sharing all works, all glory, all power, all knowledge, etc.. Giles, in singling out authority, is leading up to a claim that to assert the position of "eternal functional subordination" is to be neo-Arian.
The problem with Giles' argument is that it shares an Arian assumption. Among anti-Nicene proponents of the Fourth century a major presupposition was that if Jesus is inferior in any way, he cannot be God. The counter argument of pro-Nicenes at this point is that Jesus' apparent inferiority is distinctly related to the Economy, that is to the incarnation, crucifixion, and resurrection. Jesus is the sent Christ, the one who gives up the form of God and takes Humanity, and in his Humanity is subject to the Father.
But, the Nicene defence sets an important precedent - differentiation of role is no bar to equality of status. Christ submits to the Father but is equal and one with the Father.
Now, whether that holds in an eternal, not just an economic sense, is the debated point. But Giles' argument hinges upon the Arian presupposition, that inferiority of role cannot coincide with equality of status. This, of course, is the presupposition that many egalitarians hold - if there is real equality of status between men and women, then there cannot exist any inferiority of role. That is simply not a justifiable proposition based on the Nicene defense of the Son's self-humiliation.
I'm unconvinced by a doctrine of eternal functional subordination of the Son to the Father, but I don't think complementarianism needs such a doctrine to support itself. I do think the doctrine of the Trinity sheds important light on how we may think about status, role, and ontology, which sheds indirect light on how we think about gender roles. Without the Dispensation of the Son's self-humiliation, we could never expect humans to accept inferiority of role, without acknowledging inferiority of being. Christ makes possible submission, since it is of God to humble oneself.