This verse in 1Corinthians 11:3 by no means suggest women are of lesser value than men.
Where there is authority, there is submission. It is the Christian value of such is perfect, any discussion outside of such guideline is corrupted.
This is the way God made things. To defy it only leads to one's own peril.
Nevertheless, the Bible has cases where women were given authority. These are exceptional cases granted by God, by no means must one on one's own to play God. The apostles were all male, as well as the writers of the Biblical scrolls. Hence the completion of the Bible, God's word. This is the order of things. So that even when a woman, preaches or teaches the word of God, she must also submit to the correct interpretation of the Bible, as well as today's men.
It is especially crucial when it comes to the relationship between husband and wife.
While husband's role is to love the wife, following not his own understanding of love, but Christ's love. This is the greatest love of all. As the head, a husband needs to be hard working and initiative while also able to control his anger.
Some scholars fail to see this and thus to abide by their egalitarianism principle, they tried all kinds of twisted interpretations from cultural history to manipulation of terminology.
Women in those days were not dumb. Nor were they less educated than men. Just because one doesn't go to school doesn't make one less educated.
As for the Greek term for head: κεφαλή (kephale), it is a term used where there is submission required, no matter how one wishes to interpret it. Some Christian Scholars tried to interpret it as "source", with the intention of fearing men's status being too high above women or allowed to be abusive, which was never the case, nor was it God's will since creation. Many Biblical dictionaries (lexicography) have never interpret this term as "source".
The word "source" is better represented by "πηγη" which is used in the Bible several times as metaphor for "source" because it also means fountain.
Charles Barrett was very careful in interpreting this verse in his commentary by using resources as accurate as possible even though he has the urge to interpret the term as "source" by relating it to "origin":
"In this verse (which is to be contrasted with 4, 7, 10 below) the word head (κεφαλη) is evidently used in a transferred sense. In the Old Testament head (rosh, sometimes but by no means always translated into Greek as κεφαλη) may refer to the ruler of a community (e.g. Judges x. 18); this use, however, though it was adopted in Greek-speaking Judaism, was not a native meaning of the Greek word (for details see H. Schlier, in T.W.N.T. iii. 674 f.). In Greek usage the word, when metaphorical, may apply to the outstanding and determining part of a whole, but also to origin (e.g. in the plural, to the source of a river, as in Herodotus iv. 91). In this sense it is used theologically, as in an Orphic fragment (21a): Zeus is the head, Zeus the middle, and from Zeus all things are completed (Ζευς κεφαλη, Ζευς μεσσα, Διος δ᾽ εκ παντα τελειταιαρχη instead of κεφαλη adds to its significance; see also S. Bedale in ; that some MSS. have J.T.S. v (new series), 211-15). That this is the sense of the word here is strongly suggested by verse 8 f. Paul does not say that man is the lord (κυριος) of the woman; he says that he is the origin of her being. (p. 248)"
With such resources, Barrett failed to see that the ancient writers he was referring to have never intended that word to be used without a need of submission. Herodotus used it in plural form in a literary manner for river, not human, while Orphic fragment's indication of such term hinted a sense of supremacy. As compared to many lexicographies, this term has more often been used as headship than mere source. When the Bible uses such term, it also has the following verses to clarify this matter, not that this needs to be clarified in the first place, but the following verses and verses pertaining to 1Corinthians 11:3 are extensive requirement of the word of God as a whole.
As misleading as Barrett tried not to be, others such as Gordon Fee however took this further to completely replace the meaning of this term with "source" by rejecting the term as "head over" in his commentary.
Submission is never a bad thing. If so, Jesus wouldn't have WILLINGLY, TRULY submitted to the Father, even though He is God's equal.
Of course, this is not to say that we are God's equal when we submit to God. God is God, man is man, God is not man, man is not God. Man is created, by man, I mean woman here as well for man and woman are equal. We are however, not equal to the creator who has the authority to destroy us all. While human authorities should be exercised under God's Will.
As for examples I've personally witnessed of those male scholars who love the use of "source" for such verses in the Bible, have remarkably weak standpoint and some of those having very superficial marriage, and not able to influence or counsel others' marriages towards perfection in Christ.
If there is an abusive husband, the church must also suffer with the wife, and go through this hard time with prayer and wisdom and holiness. I'm sure there are many ways to resolve this, but whatever decision made, it must never be done to perverse God's Will.