ISSUES: The argument is undermined by it proffered support—namely—pointing at visitors and outsiders who are not church leaders, but "(Greek: idiotēs, “untrained”) and “unbelievers” (apistoi, “those without faith”) — Heiser, Michael S.. Reversing Hermon: Enoch, the Watchers, and the Forgotten Mission of Jesus Christ (p. 151). Kindle Edition.
It is hard to understand how pagan people, untrained in the doctrines of Christ and unbelievers are "messengers" whose status somehow warrants Paul going on about head coverings from God through Christ to husbands and then to wives. The argument just isn't sensible.
ISSUES: What do hairstyles have to do with divine order? Why would guardian angels care? Moreover—there is no evidence from Second Temple Era Jewish writing that any angelic being standing watch over humans cares about women's hairstyles or head covering.
ISSUES: The third argument has to do with Essene communities holding that their human worship was in a greater context of angelic worship. Thus, the cultic worship of the Essene would need women to be respectful of how the worship of people along with angels worked—keeping it all "in order". However, this view breaks down in light of non-Jewish pagan converts who don't hold to these Jewish traditions, especially in view of the removal of male circumcision as a requirement for participation in worship. Therefore, another overarching idea must be at work here for Paul to give such instructions to a Gentile church.
With the weaknesses of these three primary arguments in clear question, we must now ask: "Is there a better choice?" The answer is yes, but the details require deeper first century pagan worldview understanding to bring it together.
A more persuasive alternative
NT Scholar, Troy Martin's writeup—focused on v 13-15:
1. What is peribolaiou? (hair as a sexual function - draws up male semen)
2. Why is it a shame for a man? (hair isn't male sexual function - he is not receiving semen; therefore, male long hair seen as a "shame" or why does he have long hair?)
3. Therefore, it is nature for peribolaiou for a woman, but not for a man. Thus, the head covering is for women - "because of the angels".
This is the male side of the equation:
The female side of puberty in the first century:
The lone idea of female hair seen by these first century people as having sexual function (e.g., "testicle") doesn't fully square the text. It leaves questions.
If all women of pubescent age have physically matured enough to have sex and children, then why is Paul exclusively focused on only the married women and not all women who are sexually mature? Do unmarried sexually mature women not need protection? What could be driving this?
QUESTION: Do any of the verses from 2 to 16 point to sexually mature women in general? Yes. There are two - the first is verse 14: "Does not nature itself teach you that if a man wears long hair, it is a disgrace for him?"
While this verse primarily concerns men's appearance, it indirectly references women through the distinction drawn between men's and women's hairstyles. It does not specify if it is specifically directed at married women. However, this verse is merely clarifying the function of hair from the viewpoint of having sexual function. It does not mean that Paul's doctrine is directed at all sexually mature women. His doctrine is still pointed only at married women. Why?
The second is verse 9: "Neither was man created for woman, but woman for man."
This verse addresses the creation order and purpose of men and women, highlighting that woman was created for man. While it pertains to women in general, it again does not negate the overall target Paul has put into view: married women. Why?
Relevant notes continue here.