The covering of the face in public really isn't part of Abraham's culture. Sara herself didn't cover her face or wear a marriage covering. That is why Pharoah and Abimelech sought her until it was revealed to them she was married to a prophet. Genesis 20 reveals Abraham provided her protection. He was her covering. Men wanting a relationship with Sara were given infirmities to prove he was a prophet and she was his wife. Abraham was the covering, not cloth.
16 And unto Sarah he said, Behold, I have given thy brother a thousand pieces of silver: behold, he is to thee a covering of the eyes (mens eyes), unto all that are with thee, and with all other: thus she was reproved.
If Mohammed were a prophet equal to Abraham his wives wouldn't need coverings. Mohammed would be their covering.
The covering in this case was the marriage ornament not facial covering. Abraham was her covering. The covering of the eyes wasn't for women, but for men who might look upon Sara and covet her. Those realizing God had consequences for coveting her would move on for another beauty.
You were not supposed to look upon or lust after a married woman and Abraham was her protection. Men often covet other mens wives. The vail was a symbol like a wedding ring, people could know she was married. Those who approached Sara were stricken with calamity, which proved to Pharoah and Abimelech that Abraham was a prophet. Sara was reproved perhaps because she didn't wear a covering, she had good reason because of the danger Abraham would be killed and she taken. I believe to protect Abraham in a country he was sojourning. It was dangerous to have a beautiful wife.
Sara made a wrong decision but by God's design, since God used the infirmities it brought to prove Abraham to some kings. It solidified his connection to God and because of that he was allowed the land.
There are many other evidences from Abraham's family that the women did not normally cover their faces and in many cases did not wear a marriage covering. Rebekkah travelled with men without a vail before meeting Isaac her husband. Once in his presence she wore a marriage vail on her head, not covering the face, there is no evidence for that.
Genesis 24
65 For she had said unto the servant, What man is this that walketh in the field to meet us? And the servant had said, It is my master: therefore she took a vail, and covered herself.
She had come from Abram's brother and knew the customs, she didn't practice Islamic face covering. The marriage covering according to Paul was a symbol of authority for married women that was on the head, not the face. It likely wasn't a scarf. The point is she travelled without the vail.
She wore the vail while taken to her mother in laws tent, by her husband. Why would she hide her beauty from her husband? The muslim belief that hiding beauty from temptation as the goal, falls apart.
The marriage vail was on the head, not the face. We see this in many Catholic churches with small beaded lace coverings that don't hide beauty.
10 For this cause ought the woman to have power on her head because of the angels.
This is likely from Genesis 6 where it says the women were taken, implying the married as well. The word taken implies stolen or removed by instigation. The people who belonged to God corrupted themselves and marriage, by taking other men's wives for their own.
2 That the sons of God saw the daughters of men that they were fair; and they took them wives of all which they chose.
The symbol of authority spoken of by Paul, or marriage vail was a symbol to all men and women after the flood, a symbol that marriage was sacred. It should remind muslims as well who practice taking the spoil of non-Muslim women that God doesn't approve of taking what you choose. In Genesis 6 wearing vails wasn't practiced but was later known in Abraham's family. It wasn't face covering. Genesis 6 shows the women were open to see their beauty.
The flood came partially due to desecration of marriage, so it would be stressed after Noah. God could make all women be hidden for eternity as some sects of Islam have or simply create a symbol to remind mankind of the cause of Noah's flood. A symbol showing a woman is married and it is improper for another man to take her. Men should consider the consequences and that is a deterrence.
Those who covered the face did so to hide their identity, not to hide beauty. Face coverings protected evil doers from being identified. Harlots, taken wives, those engaged in adultery, spies, etc had reason to cover their faces.
Genesis 38
15 When Judah saw her, he thought her to be an harlot; because she had covered her face.
19 And she arose, and went away, and laid by her vail from her, and put on the garments of her widowhood.
The harlots tended to cover themselves, and laying aside the vail after deceiving Judah showed a widow of sexual age did not wear a vail publicly. She took it off once done deceiving.
The point is, Abraham's family did not wear face coverings, and even the vail wasn't law. Symbols of marriage are ok, but Islam corrupted the purpose. It wasn't to hide women. Men who have been married and know a woman's nakedness can fantasize and imagine women no matter what they wear. Skimpy clothes make it worse, but you can't stop lust by force. Muslims have as much lust or more. They aren't holier than any others.