To argue that the head covering is the proper means to show humility is to argue from a selective hermeneutic when it comes to certain passages. For example, the same apostle who commanded that women (when praying or prophesying in public, not when simply attending church) wear a head covering also said that ‘we should greet one another with a holy kiss.’ In the ancient world, as well as in Europe today, it was the men who would do this to other men. Do we follow this commandment, too? If not, why not?
There are two dangers to avoid here. First, if we fail to see the underlying principles in scripture and only look at the symbols, we might as well all join an Amish community. Second, if we assume that because the symbols may be culturally-conditioned, so also may the principles, then scripture ceases to be normative for us in any meaningful sense. I would say that the majority of Christians today have fallen into the second trap when it comes to 1 Corinthians 11.
Please understand that 1 Corinthians 11 seems to give at least four reasons that go beyond culture for a show of the woman’s submission to the elders. The principle behind the symbolism is intact; but the symbol may vary from culture to culture.