Complementarianism is not generally about authority/submission, as if they exist in a moral vacuum. It is about what biblical authority/submission look like. The passages where complementarians find reference to the authority/submission dynamic in marriage do not neglect to show us the nature of the authority.
Husbands, love your wives, as Christ loved the church and gave himself up for her, that he might sanctify her, having cleansed her by the washing of water with the word, so that he might present the church to himself in splendor, without spot or wrinkle or any such thing, that she might be holy and without blemish. In the same way husbands should love their wives as their own bodies. He who loves his wife loves himself. For no one ever hated his own flesh, but nourishes and cherishes it, just as Christ does the church… (Ephesians 5:25-29)
The husband should give to his wife her conjugal rights, and likewise the wife to her husband. For the wife does not have authority over her own body, but the husband does. Likewise the husband does not have authority over his own body, but the wife does. (1 Corinthians 7:4)
Likewise, husbands, live with your wives in an understanding way, showing honor to the woman as the weaker vessel, since they are heirs with you of the grace of life, so that your prayers may not be hindered. (1 Peter 3:7)
Husbands, love your wives, and do not be harsh with them. (Colossians 3:19)
This is what a complementarian man does.
A complementarian husband does not exercise his authority absent of any authority over him. The Bible that grounds his authority describes his responsibilities, so he submits to God's authority, and he is under the authority of a local church. Contrary to caricature, the complementarian husband who uses his views as a license to shut out his wife's counsel, to forsake her cherishing and freedom, and to deny her flourishing and edification is not a complementarian at all, but a selfish brute in need of sharp rebuke and church discipline. And if the situation calls for it (physical abuse or any other compromise of safety), legal justice.
No view of gender roles will protect us from sin, certainly. But real complementarianism is not about "lone ranger" husbands deciding according to their own whims what's good for their wives and families. It is about husbands under God's authority and the authority of a local church cherishing, honoring, and building up their families. Husbands are charged with presenting their wives sanctified and spotless to Christ, not burdened and beaten down by neglect and dishonor. A complementarian man's mission is his wife's joy in Christ. In fact, Proverbs 12:4 and 1 Corinthians 11:7 imply that a wife haggard in spirit and squelched in soul is evidence of a husband derelict of his duty, regardless of his views on gender roles. Any man who treats his wife in a neglectful or abusive way is no complementarian; he is a walking perversion of manhood in need of repentance.