According to Sarah Grimke, if Adam is the only one responsible for the fall, or sinned willingly (whereas Eve as deceived), why on earth should these be construed to give him (man) leadership over women? Dr. Payne writes on Egalitarianism and federal headship on his website. I found the following helpful (emphasis mine):
Don’t Stretch the Analogy Beyond Its Purpose
To read more into the analogy than Paul makes explicit is to risk substituting a private inference for the direct teaching of God’s Word…On p. 281 Cranfield highlights the point of the analogy: “Adam is only mentioned in order to bring out more clearly the nature of the work of Christ. The purpose of the comparison is to make clear the universal range of what Christ has done.”Other Passages Focus on Eve’s Responsibility for the Fall
Paul writes in 2 Cor 11:3, “But I am afraid that just as Eve was deceived by the serpent’s cunning, your minds may somehow be led astray from your sincere and pure devotion to Christ.” In 1 Tim 2:14-15 he writes, “the woman being thoroughly deceived and fell into transgression [lit: ‘has become in transgression’], but she will be saved through the Childbirth if they remain in faith and love and holiness with propriety [to counteract socially unacceptable behavior here just as in 1 Tim 2:9 (cf. Cor 11:3–16)].” The association of Eve with her descendents, as the equation of “she” with “they” requires, shows that her transgression affects them, and that her seed (Gen 3:15) crushes the serpent, Satan, and brings salvation to them through “the Childbirth.”Genesis Holds Both Responsible
Since in Hebrew “Adam” identifies both the man in isolation and the man and the woman together in Gen 1:26-27, “let us make man in our image, in our likeness, and let them rule…,” and since it was the sin of both the woman and the man that led to the fall in Gen 3, and since God addresses both woman and man (in that order) stating the consequences of the fall, and since there is a corresponding plural contrasting God as “us” and man as “them,” it seems inescapable that Eve did participate in the fall and in its ultimate reversal through her seed.“Mankind” not “Male”
One possibility is that Paul intended “one man” to refer to the Adam (singular) identified as “them” in Gen 1:26-27, as possibly indicated by Paul’s use of ἄνθρωπος rather than ἀνήρ. Paul may have chosen to use the word “Adam” because it is a singular name, even when used to designate the original couple, and so makes a more direct counterpart for the singular name, the one man (ἄνθρωπος, focusing on his humanity, not his being male), Jesus Christ in 5:15, who brings the opposite, salvation from that fall…It is not Adam or Christ as male that is highlighted here, but Christ as human, hence the use of ἄνθρωπος in “though one human being” rather than ἀνήρ, which Paul could have used if he had intended to specify man as male.Payne also cautions that neither the terms “federal” or “headship” are used in this portion of Scripture concerning Adam. These are terms some of us are using to describe what we think the text means, not what the text says exactly. I would put the argument on par with universalism claiming their interpretation of the text.