DMRF, you do not sound legalistic in your question at all. Digging into scripture on the matter is the thing to do. Because what is said in The Bible is what we must test all things to...because our human understanding can get us into making assumptions that just don't add up to what scripture and the spirit say, together.
It is most often our preconceived ideas and indoctrinations of man that cause us to do things contrary to what the Bible actually says. We must never cherry pick scripture to fit what we want to believe, but look at the whole and what it says. I have found when we do that, a picture begins to form that brings clarity to The Word.
Firstly, I will address the OP scripture used, then share a link to scripture regarding "above reproach" In 1 Corinthians 11, Paul begins by speaking of 'coverings' as they pertain to the household of God. At the end of this passage, he says the woman's hair is her glory/covering.
If we are of the household of God, it is very important to grasp this truth.
(Someone else has also shared scripture about a man is to dress like a man, etc.)
I am sharing from the Amplified version.
3 But I want you to understand that Christ is the head (authority over) of every man, and man is the head of woman, and God is the head of Christ.
4 Every man who prays or prophesies with
something on his head dishonors his head [and the One who is his head].
5 And every woman who prays or prophesies when she has her head uncovered disgraces her head; for she is one and the same as the woman whose head is shaved [in disgrace].
6 If a woman does not cover her head, she should have her hair cut off; and if it is disgraceful for a woman to have her hair cut off or her head shaved, she should cover her head.
7 A man ought not have his head covered [during worship], since he is the image and [reflected] glory of God; but the woman is [the expression of] man’s glory.
8 For man does not originate from woman, but woman from man;
9 for indeed man was not created for the sake of woman, but woman for the sake of man.
10 Therefore the woman ought to have a
sign of authority on her head, for the sake of the angels [so as not to offend them].
11 Nevertheless, woman is not independent of man, nor is man independent of woman.
12 For as the woman originates from the man, so also man is born through the woman; and all things [whether male or female] originate from God [as their Creator].
13 Judge for yourselves; is it proper for a woman to offer prayer to God [publicly] with her head uncovered?
14 Does not common sense itself teach you that if a man has long hair, it is a dishonor to him,
15 but if a woman has long hair, it is her ornament
and glory? For her long hair is given to her as a covering.
16 Now if anyone is inclined to be contentious [about this], we have no other practice [in worship than this], nor do the churches of God [in general].
Here are lots of scripture to dig into:
What does the Bible say about "Above Reproach"
What Does the Bible Say About Above Reproach?
What does the Bible say about Church Leadership/ministry
What Does the Bible Say About Church Leadership?