I understand what you're saying, and it's a fair challenge to make. But there is another major thing to explain that I ran out of time to earlier, which is a radically different way of looking at the entire issue, and I think draws together the covering vs hair issue into a single clear narrative. It sounds shocking at first to a Western reader, but read to the end, it all makes sense...
I don't think this is referring to a "hair covering", but actually a "head covering". The shawls etc that are commonly called "head coverings" by Christians do not actually cover the head. They only cover the hair. The face is still visible, so the head has not been covered. To truly cover the "head" requires the face to be veiled.
In 1 Corinthians 11, consider the following key words: But every woman that prayeth or prophesieth with her head (G2776) uncovered dishonoureth her head: for that is even all one as if she were shaven. For if the woman be not covered (G2619), let her also be shorn ... But if a woman have long hair, it is a glory to her, for her hair is given her for a covering (G4108).
G2776: kephale. Means "the head".
G2619: katakalupto. Thayers: 1. To cover up, 2. to veil or cover one's self. Strongs: To cover wholly, that is, veil: - cover, hide. (emphasis in original)
G4108: peribolaion: Thayers: A covering thrown around, a wrapper. a. a mantle, b. a veil. Strongs: Something thrown around one, that is a mantle, veil: - covering, vesture. (emphasis in original)
This is not talking about a headscarf or a hat. It is talking about a veil - something that wholly covers the entire head, that fully veils the face as well as the rest of the head.
These days we'd mostly associate that with Islam - but even our own fashion retains the veil at least symbolically. Women wear veils at weddings, older women still sometimes wear veils at funerals. Look at a load of women done up fancy to go to the races, or upper-class people like the British royal family attending a formal event, and you'll see plenty of veils - mesh veils that don't really cover anything, but still veils. This is part of our heritage, it's not weird.
And it makes sense of the hair. If I rewrite 1 Corinthians 11 from the KJV substituting the word "veil" for "covering", the whole thing takes on a new, but entirely consistent, meaning. That's not a fringe reading - the ASV, CJB, NRSV and RSV all use the word "veil" through most of this passage, translating "katakalupto" as "veiled", however here I have also rendered "peribolaion" as "veil" also, which in my mind ties the entire thing together better and is clearer.
But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.
Every man praying or prophesying, having his head veiled, dishonoureth his head.
But every woman that prayeth or prophesieth with her head unveiled dishonoureth her head: for that is even all one as if she were shaven.
For if the woman be not veiled, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be veiled.
For a man indeed ought not to veil his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.
...
Judge in yourselves: is it comely that a woman pray unto God unveiled?
Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him?
But if a woman have long hair, it is a glory to her: for her hair is given her for a veil.
Now the hair makes sense - and why it is important that her hair is "long"! A woman is supposed to veil her head when praying. But if she has long hair, it already covers most of her head, and it is given to her as a veil! All she has to do is pull it over her face, or simply lean forward so it falls over her face, and she's veiled! Isn't God wonderful providing her everything she needs! But if she refuses to veil herself with a cloth, or the hair God has given her, then she may as well have her hair cut short, since she refuses to use it for its God-given purpose then she doesn't deserve to have it.
This also makes 1 Corinthians 11 fit in with the rest of scripture. We don't have any other instructions in scripture for women to use head coverings. But there are a number of references to veils (e.g. Rebecca veiling herself when meeting Isaac). If this is about veils, then there are multiple witnesses, and it is no longer an isolated passage.
This has some rather interesting implications. If a woman has her hair tied up in a headscarf so it is out of her way, and cannot fall over her face, is she truly covering her head? Or is she actually disobeying the command to veil herself? In following human tradition on what the "head covering" refers to is she actually ending up in disobedience to the very word of God on the matter?
So back to my own practice - I don't think this means my women should wear what Western conservative Christians call a "head covering". I think requiring that would actually be to distract them from the true meaning of this. If anything, I believe it means they should be choosing to veil their face when praying. If using their hair - that is a behaviour that is something a woman chooses to do, or not do, each time she prays. It is a matter of personal behaviour, a personal decision on how to humble herself before her God, it's not a matter of clothing for me to require. If it means she should truly wear a cloth "head covering" in public - well I'd have to be a lot more convinced that I was correct and this was critically important to start requiring my women to wear a niqab.
So I require my wife and daughters to have long hair. That is all. But I welcome any further coverings they choose to add to that out of their own conviction.