I do see what you are thinking towards the greek word for angels. Let me explain.
According to the Christian New Testament, angels today observe the behavior of men on earth (1 Cor 4:9; Eph 3:10; 1 Tim 3:16; and 1 Pet 1:12). They are not able to read minds to the contrary. "True, the angels cannot penetrate into the inward devotion of the mind, which God only observes; but they observe and take notice of the outward decency of our carriage, and the reverence of our deportment." - Burkitt's Commentary on 1 Corinthians 11:10
Next we need to recognize that only modern liberal (in most cases) scholarship has argued that it is messenger and not angels. All others in Church history pre-demoinationly held to this view of angels.
The greek Septuagint uses the word angels, all early Church fathers who talk about this mention angels. Most commentaries and denominations hold to the view it is angels. It is only in the last 50 years or perhaps 100 years some theologians have started to try and take apart some of these practices, especially in light of feminism.
It so happens that the Septuagint version has here the word κατακαλυπτων, which Paul has been using. So perhaps Paul’s reference to the angels is meant to recall how the seraphs covered themselves, in which case the idea would be that if the angels themselves do this, how much more should a woman.
It is simply indisputable by greek scholars and early Church Fathers and Christian Churches that almost all if not all people took it to mean angels. The only dispute primarily was if they were good or bad angels or both.
Here is a listing of about 20 commentaries that touch on this and a few quotes, this list could be 1000 times longer as again everyone believed this doctrine and practice in the Church till only recently, trust this helps brothers and sisters:
“The reason why our sisters appear in the House of God with their heads covered is ‘because of the angels.’ The apostle says that a woman is to have a covering upon her head because of the angels, since the angels are present in the assembly and they mark every act of indecorum, and therefore everything is to be conducted with decency and order in the presence of the angelic spirits.” Charles Spurgeon – Spurgeon’s Sermons on Angels, (Kregel Academic, 1996), page 98.
“Paul’s admonition for women to wear a head covering ‘because of the angels’ removes any doubt that this teaching is universal and timeless.” – K.P. Yohannan – Head Coverings (2011, Believers Church Publications) Page 24
because of the angels—who are present at our Christian assemblies (compare Ps 138:1, "gods," that is, angels), and delight in the orderly subordination of the several ranks of God's worshippers in their respective places, the outward demeanor and dress of the latter being indicative of that inward humility which angels know to be most pleasing to their common Lord (1Co 4:9; Eph 3:10; Ec 5:6). Hammond quotes Chrysostom, "Thou standest with angels; thou singest with them; thou hymnest with them; and yet dost thou stand laughing?" Bengel explains, "As the angels are in relation to God, so the woman is in relation to man. God's face is uncovered; angels in His presence are veiled (Isa 6:2). Man's face is uncovered; woman in His presence is to be veiled. For her not to be so, would, by its indecorousness, offend the angels (Mt 18:10, 31). She, by her weakness, especially needs their ministry; she ought, therefore, to be the more careful not to offend them." - Jamieson-Fausset-Brown Bible Commentary on 1 Corinthians 11:10
"The most natural interpretation seems to me to be this: "A woman in the public assemblies, and in speaking in the presence of people, should wear a veil - the usual symbol of modesty and subordination - because the angels of God are witnesses of your public worship Hebrews 1:13, and because they know and appreciate the propriety of subordination and order in public assemblies." - Barnes' Notes on 1 Corinthians 11:10
"διὰ τοὺς ἀγγέλους] On account of the angels: i.e. because in the Christian assemblies the holy angels of God are present, and delighting in the due order and subordination of the ranks of God’s servants,—and by a violation of that order we should be giving offence to them." - Greek Testament Critical Exegetical Commentary
"The simplest explanation (since Paul was speaking of the proper subordination of woman) is that this is a reminder that the "angels who kept not their first estate" lost heaven; and it is not far-fetched to draw the analogy that those precious angels called women should not go beyond the limitations imposed upon them by their creation." - Coffman's Commentaries on the Bible
" It is best on the whole to regard it as an intimation that the angels, though invisible, were fellow-worshippers with men in the Christian assemblies, and were therefore ‘spectators of the indecency,’ and liable to be offended thereat. ‘When therefore the women usurp the symbol of dominion, against what is right and lawful, they make their shameful conduct conspicuous’ in the eyes of the messengers of God. Thus Calvin. Erasmus paraphrases it well: ‘If a woman has arrived at that pitch of shamelessness that she does not fear the eyes of men, let her at least cover her head on account of the angels, who are present at your assemblies.’ " - Cambridge Greek Testament for Schools and Colleges on 1 Corinthians 11:10
"For this cause also a woman ought to be veiled in the public assemblies, because of the angels - Who attend there, and before whom they should be careful not to do anything indecent or irregular." - Wesley's Notes on 1 Corinthians 11:10
"Because of the angels (δια τους αγγελους — dia tous aggelous). This startling phrase has caused all kinds of conjecture which may be dismissed. It is not preachers that Paul has in mind, nor evil angels who could be tempted (Genesis 6:1.), but angels present in worship (cf. 1 Corinthians 4:9; Psalm 138:1) who would be shocked at the conduct of the women since the angels themselves veil their faces before Jehovah (Isaiah 6:2)." - Robertson's Word Pictures in the New Testament
"The holy angels, who were supposed by both the Jewish and the early Christian Church to be present in worshipping assemblies. More, however, seems to be meant than “to avoid exciting disapproval among them.” The key-note of Paul's thought is subordination according to the original divine order. Woman best asserts her spiritual equality before God, not by unsexing herself, but by recognizing her true position and fulfilling its claims, even as do the angels, who are ministering as well as worshipping spirits (Hebrews 1:4). She is to fall in obediently with that divine economy of which she forms a part with the angels, and not to break the divine harmony, which especially asserts itself in worship, where the angelic ministers mingle with the earthly worshippers; nor to ignore the example of the holy ones who keep their first estate, and serve in the heavenly sanctuary." - Vincent's Word Studies
"The most and best interpreters understand here by angels, the good angels; for the apostle would hardly have spoken of devils under the notion of angels, especially speaking to deter persons from actions; and so it teaches us, that the good angels, who are ministering spirits for the good of God’s elect, at all times have a special minstration, or at least are more particularly present, in the assemblies of people for religious worship, observing the persons, carriage, and demeanour; the sense of which ought to awe all persons attending those services, from any indecent and unworthy behaviour." - Poole's English Annotations on the Holy Bible on 1 Corinthians 11:10
"When, therefore, women venture upon such liberties, as to usurp for themselves the token of authority, they make their baseness manifest to the angels. This, therefore, was said by way of amplifying, as if he had said, “If women uncover their heads, not only Christ, but all the angels too, will be witnesses of the outrage.” And this interpretation suits well with the Apostle’s design. He is treating here of different ranks. Now he says that, when women assume a higher place than becomes them, they gain this by it — that they discover their impudence in the view of the angels of heaven." - Calvin's Commentaries on the Bible on 1 Corinthians 11:10
"angels i.e. of the presence of the angels." - Scofield's Reference Notes on 1 Corinthians 11:10
"the universal tradition among the Jews that the angels fell by lust for mortal women, which was the Jewish way of interpreting Genesis 6:1, 2. This is the view of Tertullian ('De Virg. Vel.,' 7) in writing on this subject. A woman, in the opinion and traditions of Oriental Jews, is liable to injury from the shedim, if she appears in public unveiled; and these evil spirits are supposed to delight in the appearance of unveiled women... On the whole, however, the meaning seems to be, out of respect and reverence for the holy angels, who are always invisibly present in the Christian assemblies." - Pulpit Commentary on 1 Corinthians 11:10
"o understand them of good angels, who attend the assemblies of the saints, and observe the air and behaviour of the worshippers; wherefore women should cover their heads with respect to them, and not give offence to those pure spirits, by an indecent appearance: it is agreeable to the notions of the Jews, that angels attend public prayers, and at the expounding of the word; they often speak (f) of an angel, "that is appointed over prayers"... Moreover, this veiling of the woman in public worship because of angels, may be an imitation of the good angels, who when they sung the praises of God, and adored and glorified his perfections, covered their faces and their feet with their wings, Isaiah 6:1." - Gill's Exposition of the Entire Bible on 1 Corinthians 11:10
"She ought to have power, that is, a veil, on her head, because of the angels. Their presence should keep Christians from all that is wrong while in the worship of God. Nevertheless, the man and the woman were made for one another. They were to be mutual comforts and blessings, not one a slave, and the other a tyrant. God has so settled matters, both in the kingdom of providence and that of grace, that the authority and subjection of each party should be for mutual help and benefit. It was the common usage of the churches, for women to appear in public assemblies, and join in public worship, veiled; and it was right that they should do so." - Matthew Henry's Concise Commentary on 1 Corinthians 11:10
"But what are we to make of “the angels,” for whose sake woman is not to put off this power? Now closely following the passage in Genesis to which Paul refers, there is the story of the first infraction of the true relation of the sexes (Genesis 6:1-4), which the rabbis read thus:--The daughters of men, departing from their primitive simplicity and decorum, laid aside their veils, and tricked out their hair and faces with ornaments. The angels saw them, and grew enamoured of their beauty, and fell from their blessedness." - The Biblical Illustrator
"All these senses the learned bishop rejects, and believes that the apostle uses the word angels, in its most obvious sense, for the heavenly angels; and that he speaks according to the notion which then prevailed among Jews, that the holy angels interested themselves in the affairs of men, and particularly were present in their religious assemblies, as the cherubim, their representation, were present in their temple. Thus we read in Ecclesiastes 5:6 : Neither say thou before the Angel, it was an error; and in 1 Timothy 5:21 : I charge thee before God and the Lord Jesus Christ, and the elect Angels, etc." - Clarke's Commentary on 1 Corinthians 11:10
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