It is of immense importance if you are a woman. Whatever Christ is to the husband, the husband is to the woman. It would be good to know your husband is your conduit to Christ or if Christ is personally available directly to you. If you read head as anything other than source, it would almost knock out any direct relationship between the woman and Christ. As we know, that would contradict the rest of the Bible. We tend to gloss over what the words actually say in that verse because we think we already know what it reads so we don't actually read it.
Further, the change of man to husband then leaves out single women or widows. This leads to a man-made embellishment of either the pastor or their father being the head in order to make the theology work that people try to pull out of this verse. I think anyone who can put their preconceptions to the side and reread the verses for exactly what it says would have to conclude that it can't be speaking of any kind of authority order.
The change of word didn't cause it; the change of word is merely one subtle way it has been interjected. The the eternal subordination of the Son (ESS) heresy is a far larger than I could deal with in a simple post but you can google Grudem and Ware's doctrines on this (btw it has infiltrated much of the seminary schools by way of inclusion in Grudem's Systematic Theology textbook). They believe in complimentarianism to such an extent that they see it in everything, including the Trinity and it affects their word choices. They use this verse to claim that Christ is eternally subordinate to the Father because they change man/woman to husband/wife which leaves one to only be able to render the word "head" as authority (authority is a 2nd meaning to head in English but not in Greek), and then allows them to further construct the ESS heresy.
I was willing to believe that until I studied up on this group of people, and learned that their doctrines were informing their Bible rather than vice versa.
Actually I did do so for many years until I was led to study the passage in greater detail. After about 2 years of studying many resources I came to realize the passage's message really didn't have a lot to do with hats. If you read it carefully, the passage talks about a woman having a cover, but then in v 15 it seems to say "never mind all that! A woman's hair is her cover."
The word for cover in the Greek that is used throughout the passage can be used for either cover or hair, but the last use of cover in v. 15 is from a different Greek word that only be used for a cloth cover, where Paul says that a woman's long hair is her cover:
1 Cor 11:15: 15 But if a woman have long hair [2863], it is a glory to her: for her
[long or ornamental] hair [2864] is given her for a covering [4018].
So, the word "cover" is defined in v 15 as long hair. The word used for hair here is a Greek word that indicates long hair. So if you read the passage using Paul's definition "long hair" instead of the term "cover" it will make more sense.
STRONGS NT 2863: κομάω - Usage: I wear the hair long, allow the hair to grow out.
κομάω, κόμω; (κόμη); to let the hair grow, have long hair (cf. κόμη at the end):
1 Corinthians 11:14f (In Greek writings from Homer down.)
STRONGS NT 2864: κόμη - Usage: hair, long hair.
κόμη, κόμης, ἡ (from Homer down), hair, head of hair:
1 Corinthians 11:15. (According to Schmidt (21, 2) it differs from θρίξ (the anatomical or physical term) by designating the hair as an ornament (the notion of length being only secondary and suggested).
STRONGS NT 4018: περιβόλαιον - Usage: a wrapper, mantle, veil, cloak, covering.
περιβόλαιον, περιβολαίου, τό (περιβάλλω), properly, a covering thrown around, a wrapper; in the N. T.
The passage actually had to do with not behaving in an unseemly way or bringing reputational shame upon the worship of Christians. The problem Paul was addressing was that the people of Corinth were heavily involved in worshiping a goddess. The goings on in her temple were highly sexual, orgiastic included public displays of homosexual behavior and involved men wearing effeminate, ornamental hair like women.
Paul was exhorting the Christians to appear normal to their gender and not give anyone cause to think Christian worship was anything like the 'worship' that was going on at the goddess's temple.
As to the passage about women being shaven, at the time, a woman who was accused of adultery was shaven (shorn) to publicize the shame she was accused of.
That's fine, but you have other verses to draw from if that's your perspective; you don't need this verse to prove it. I'm not trying to make this a complimentarian discussion. That's a whole different discussion and on these forums people get pretty heated and offensive so those threads usually get shut down. We were only discussing your initial question about whether there is any passage in the ESV where the translation makes a difference so I called out one of them, 1 Cor 11:3.