Women and Ministry: Headship from Home to Church?

In chp 2 of RT France, Women in the Church’s Ministry, the subject of authority is addressed. France contends that at the bottom of the discussion about the role of women in ministry among evangelicals in the Anglican communion was the issue of authority.

Is the issue, at bottom, one of authority — preservation of authority, fear of switching authority, etc? Do you think this is why most are opposed to women in ministry? (I originally had a slightly longer version of this post; I’ve reduced it.)
The singular issue he probes here is what the NT says about “headship” and whether or not such headship, which is clearly taught for the home (whatever “headship” means), shifts from home relations to social and church relations. In other words, do the NT writers appeal to “headship” issues when it comes to discerning what women can do in ministry? (France argues, and I anticipate here, probably not.) Here are his conclusions:
First, there is a sense of “order” in the NT concept of “submission” — that is, it means to live in the order established, but he does not leave this sense there. Hence, order is linked to a “call for mutual responsibility and indeed mutual submission” (47). He points to 1 Corinthians 11:11-12 and Ephesians 5:21.
Second, whatever headship means at home, that category is not used to discern the activities of women in public.
Third, when “submit” is used in the context of ministry/church — which occurs only in 1 Cor 14:34-35 and 1 Tim 2:11-12, the first refers to “husbands and wives” and probably not men and women in general. The second could refer to husbands and wives, but may not mean that. Some today — perhaps many — think both passages are about husbands and wives (not men and women) and their relationship in the local church — and many would also add to this that cultural expectations both influenced Paul and have changed.
Fourth, 1 Cor 11:2-16 (head covering passage) is about how women dress in public (church) and not about leadership and ministry. What is most notable about 1 Cor 11 is that Paul clearly permits a woman to exercise the gift of prophecy in public — which entails Spirit-inspiration and authority.
Thus, 1 Cor 11 according to France actually moves against the view that 1 Cor 11 restricts ministry; it actually permits ministry.
Hermeneutically … France asks (but does not answer in this chp) if the “order” of these passages is transcultural and if these passages apply in the same way to an egalitarian worldview? What are your thoughts?