The first thing to notice about Paul's admonition against women taking leadership in worship is that it limits or clarifies what Paul just said about all Christians prophesying (clarifying and explaining Scripture). Here he says that women should keep silent at church during worship. (This may be at odds with 1 Corinthians 11:5, which implies that women may pray or prophesy as long as their heads are covered. The issue may be that they are not to do so at church or during worship, but that other venues may be okay. The issue may also be one of permission, in that the issue is headship and submission.) In 1 Corinthians 14:34 Paul links the submission of women to Genesis 3:16, "Your desire shall be for your husband, and he shall rule over you."
John Gill (1697 - 1771) notes, "By this the apostle would signify, that the reason why women are not to speak in the church, or to preach and teach publicly, or be concerned in the ministerial function, is, because this is an act of power, and authority; of rule and government, and so contrary to that subjection which God in his law requires of women unto men. The extraordinary instances of Deborah, Huldah, and Anna, must not be drawn into a rule or example in such cases. ... All speaking is not prohibited; they might speak their experiences to the church, or give an account of the work of God upon their souls; they might speak to one another in psalms, hymns, and spiritual songs; or speak as an evidence in any case at a church meeting; but not in such sort, as carried in it direction, instruction, government, and authority. It was not allowed by God that they should speak in any authoritative manner in the church."
Worship leadership was given to men, not because they were better at it, nor because they enjoyed a superior status. Rather, I suspect that it was a function of need. It is for the sanctification of the men. How so?
We know that the leader always gets more out of the lesson than the student, if only because the leader has spent more time in preparation. And because men are naturally lazy and irresponsible, God chose them to lead worship because they need to spend more time in preparation, because they need the additional burden of leadership responsibility to counteract their natural tendency to be irresponsible, and because they need the additional sanctification that results from additional effort.
Women tend to suffer from a different character flaw. Women, following Eve's lead (Genesis 3), tend to usurp authority. Women, because they are the natural caretakers of children, have a natural tendency to take charge of everything. That tendency, coupled with the character flaw of men toward irresponsibility, is socially and culturally deadly because it encourages the growth of male irresponsibility. Thus, God requires each sex to counteract their natural tendencies. Women are to submit rather than to take charge, and men are to be responsible rather than irresponsible. Each sex is required to grow in sanctification by denying its natural tendency and engaging its weakness -- submission for women and responsibility for men -- in order to grow spiritually. Clearly, team effort, mutual support and mutual encouragement are required. Men and women are mutually dependent upon each other for their spiritual growth.
The real strength of Christianity is not so much in Christian leadership, as in Christian followership. All Christians are followers. Men are to model Christian followership outside the home through submission to church and civil authority, and to model representative leadership in the home by representing the authority of Christ to their families. Women are to model Christian followership inside the home through submission to their husbands (who are to be in submission to Christ), and to model representative leadership by representing the authority of her husband to her children. Representative authority is always a chain of command. And to break any link of the chain is to break the whole chain.
Paul goes on to say that women should look to their husbands for spiritual instruction and leadership as an expression and exercise of biblical social order. Why? Because every Christian husband serves as a priest to his family. John wrote in Revelation, "And they sang a new song, saying, 'Worthy are you to take the scroll and to open its seals, for you were slain, and by your blood you ransomed people for God from every tribe and language and people and nation, and you have made them a kingdom and priests to our God, and they shall reign on the earth'" (Revelation 5:9-10).
Christians are to become a kingdom of priests. Paul had written earlier that "the head of every man is Christ, the head of a wife is her husband" (1 Corinthians 11:3). Thus, the best thing a wife can do for the spiritual growth of her husband is to look to him for spiritual leadership by taking her questions to him, to model submission to him as a way to encourage his growth in responsibility and counteract his tendency toward irresponsibility. The best thing a wife can do for the spiritual growth of her husband is to teach the importance and structure of biblical authority to her children and to the society at large by modeling submission to his representative authority. However, submission is not just for women, men also need to model submission to Christ. Indeed, no issue is more important, more neglected or more needed than the this issue of living in submission under representative authority.
Consequently, the reason that "it is shameful for a woman to speak in church" (1 Corinthians 14:35) is that it undermines the authority of God's intended social structure for human society -- God's representatives -- and leads to anarchy -- not right away, of course, but eventually. Because all authority belongs to God, the undermining or ignoring of any authority leads to the eventual destruction of authority, and ultimately to anarchy, the denial of authority.