It is indeed a difficult passage. I for one find the argument of those who claim it was just a cultural requirement of modesty very unconvincing, since Sacred Scripture instead presents two reasons for it: the relationship between men and women and the angles.
The thing about the angels is not explained at all. I once read a footnote that said St. Paul may be suggesting angels will punish those who disobey this law. That’s as good an explanation as any other I can think of.
Regarding the relationship between men and women, I’d think of it this way. In human relationships there are both horizontal and vertical dimensions, vaguely reflecting the horizontal and vertical relationships between the Persons of the Holy Trinity. Today, in this post-Enlightenment world, it’s fashionable to focus on the horizontal relationships, and it is indeed good to keep them in mind. For example, men and women are equally made in the image of God, and are of equal inherent human dignity. This is analogous to the equal divinity of the Persons of the Trinity.
On the other hand there is also a vertical, hierarchical relationship between people, reflecting the unequal relationships between the Persons of the Trinity. The Father begets the Son not the Son the Father, and the Holy Spirit proceeds from the Father and (or through) the Son, not any other combination. Similarly Eve came from Adam not Adam from Eve, and Cain, Abel, and Seth came from Adam and Eve not Adam or Eve from one of their children. These relationships are reflected even in the later structure of the family. Husbands and wives are of equal human dignity, yet the wife is subject to the husband. Children and parents are of equal human dignity, yet children are subject to their parents.
In healthier, less fractured societies there was less division of the nuclear family from the extended family, the tribe, and the nation, and so the relationships within the family were extended to the wider society. Women therefore were subject to men and children to adults. When the Church was formed it functioned as a sort of big extended family or multicultural nation, and the same basic dynamics between people of different sexes and ages still applied, albeit in a spirit of mutual love and respect which served to disproportionately attract women to the Church and caused some early Christian writers to suggest an ideal of treating men and women without distinction.
This is the context of St. Paul’s admonition. In confronting an almost anarchical Corinthian Church he’s trying to reinstate law and order, including by insisting on women wearing a veil as a symbol of their obedience to their husbands and the clergy, and by extension to men in general. This shouldn’t be interpreted as in any way contradicting the equal dignity of women and Christianity’s particular call to uphold that dignity. We should keep both the horizontal and vertical dimensions of the relationship together, not pit one against the other.
Today the veil has become entirely optional, but that doesn’t mean the underlying concepts expressed in scripture have lost their truth.
That’s my thoughts on the subject at the present, anyway.