Don't read into v. 3.
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.
Authority and submission are in play all over the place. God has it be that way. Christ is the head of every man, not just married ones. And God is the Head of Christ. Have a problem with that? Didn't think so. That's good with you. But what about the one sandwiched between those two? The man is the head of the woman.
I'm going to assume that you don't think that "the man" and "the woman" are a universal, invisible man or woman, but representative of the particular. But now the spin begins. And I'm not going to talk about what that is, just that men in general are in headship over women in general. Men are in authority and women are in submission. So 1 Corinthians 11 isn't talking about marriage, that married women should submit to their husbands. No, women are to submit to men, and that should show up everywhere. If you are arguing, move down to v. 16, because there shouldn't be any contention over this. The apostles weren't arguing over it, so no one else should have been.
Maybe you're good with all this so far, but I'm guessing it's already controversial, just like it would have been in Corinth---hence, v. 16. But the point of the post is about the symbol of male authority in dress and then the symbol of female submission in dress. Does God expect or assume that from His own? Yes, He does. 1 Corinthians 6:10.
Paul makes his point about the female head-covering in vv. 6-7 and then argues for it in vv. 8-10. In v. 10, he writes:
For this cause ought the woman to have power on her head because of the angels.
Paul's argument is about angels. I'm not even going to go into that. I want you to consider just the one point.
A woman ought to have power on her head. Power on her head? What's that? I'm not endorsing the ESV one iota, but here is how the ESV guys translated the same Greek words. I believe they are correct in their understanding of what that is, when they translate, "a symbol of authority on her head."
The head-covering in Corinth was a symbol of the man's authority over the woman, whether it be a father and a daughter, a husband and a wife, or whatever other man-woman relationship it was. It wasn't to "tell the difference between men and women." It wasn't to "make the man look more masculine and the woman to look more feminine." It was a symbol of subordination for the woman. Women were not wearing it because they knew what it symbolized. That was the whole point to not wearing it. Women weren't abstaining from the veil because they wanted to look like men, but because the veil meant submission.
Paul spent all of these verses about this one issue, what some want to dismiss as "non-essential." If it was non-essential, then Paul could have just said, be charitable toward the women not wearing the head-covering, because in non-essentials after all, there is charity, and in all things unity. The modern idea of unity is accepting the no head-covering wearing women. They're fine.
In Corinth, the head-covering was one symbol of female subordination of women to men even as not wearing it was a symbol of male headship of men over women. We still do have a symbol of male authority and female submission in our culture. For men it is pants and for women it is the skirt or dress. This symbol still does exist, although not to the extent that it once did, but that is in a major way because professing Christians have conformed to the world. Whatever the symbol is in the culture, that is the one that Christians should follow. If the world has no symbol, Christians should originate their own. They don't need to, however, in the United States, because we already have the symbols here. The symbols, however, are necessary because of, among other reasons, created order (v. 9) and because of the angels (v. 10). If anyone understands authority and submission it is angels. One insubordination landed angels under God's condemnation forever.
I know some people believe that Paul made the head-covering permanent. I'm not going to argue about that in this post. I know some will say that wearing the veil is only for worship time, during praying and prophesying. I know that others will say that the symbol is nothing more than the hair length. I'm not going to debate any of that. These verses cannot be nothing! Most evangelicals and fundamentalists interpret them the same way that I do in this post. They simply choose to disobey them, to rebel against them, to snub their noses at God. And then they provide cover for each other by mocking those who call for obedience. Or they attempt to muddle the discussion to the degree that, since no one can actually know what these verses are talking about, they don't have to practice them. That's not going to fool God. Not. at. all.