Tom Huguenot says:
This sums up what seems to be going on with a lot of complementarians. Complementarians are Protestants committed to sola scripture and some of them just stick with a few basic tenants for liturgy and ecclesiology. Others are just patriarchalists in the closet ... or not so much in the closet as recognizing the public relations campaign is better served by using a different label. As Walter Martin used to say, cults are groups of Christians who employ a different meaning for a term than what a normal Christian might use. A complementarian that keeps finding reasons women have to be removed from more and more roles within the church is not necessarily a complementarian.
Long ago women weren't allowed to have public roles in Western church worship. Couldn't speak, weren't even singing in front of congregants. Back then there was this high church tradition of castrating boys who had good singing voices in the hopes of forestalling their puberty altogether and keeping the golden tones. Most of the time this failed and then church doctrine and teaching held that these castrated boys couldn't get married because they were unable to produce offspring. But in its way it was a consistent outworking of precluding women from being public participants or even potential leaders in liturgy. Contemporary "complementarian" advocates of patriarchy may not recognize in themselves a comparable impulse. You follow the arguments some people have far enough there's a risk of returning to the place of creating castrati. There's nothing new under the sun, after all. Of course the production of castrati wouldn't happen right away, it would take several generations and necessitate a more successful set of cultural re-engineering on the part of some folks with faintly post-millenial sympathies.
If the sorts of complementarians who say women should not be reading scripture publicly come to this conclusion on the basis of saying that reading scripture is to hold a teaching position or position of authority then should they not retroactively insist that Mary did not sing the Magnificat? As an unwed teenage girl she lacked the spiritual authority to teach men and so the song attributed to her must have been a pious forgery on the part of the evangelist. The Magnificat shows various points of being indebted to Hannah's song in 1 Samuel but let's not imagine that Mary would have known of that being a peasant girl so perhaps both Hannah's song and Mary's song were composed retroactively on their behalf. It wouldn't be proper, given the precept that a woman should not read Scripture in a position of potentially instructing men, that women should have been used by the Holy Spirit to actually create scripture, would it? Well, time to bring back the castrati, I guess.
Or we could consider that there are some basic flaws in reasoning with this whole approach that says that women should not be permitted to read scripture in churches. A 'testimony' vs 'teaching' distinction is not a useful distinction because the psalmists wrote songs of testimony about the goodness of Yahweh that were later cited by apostles as being prophetic works. If a vexed housewife like Hannah composes a song of thanks to the Lord, well, what can a complementarian who wants no women speaking authoritatively over men to do about that? What do we imagine prophets were doing when they prophesied? They couldn't have been testifying to the greatness of God, could they? They were teaching and, well, we can't have women doing that. We'll just pretend Deborah and Huldah don't count. Mary can't count, either. The daughters of Phillip the evangelist wouldn't have been given the ability to prophesy because that would be a violation of creation order.
As Tom Huguenot put it above, it can seem as though a lot of what passes for "complementarian" commentary on gender roles and liturgy is not a defense of a tradition or church polity approach so much as a meta-commentary on the changing scope and role of gender in contemporary society.