Not sure how you proved anything. Your translation is not correct.
"And tell the believing women to lower their gaze and to guard their private parts and not to show their adornments except that of it which normally shows. They shall cover their cleavage with their khimar. They shall not show their adornments except in the presence of their husbands, their fathers, the fathers of their husbands, their sons, the sons of their husbands, their brothers, the sons of their brothers, the sons of their sisters, other women, their slaves, the male attendants who have no sexual desire and the children who are yet to attain awareness of women’s nakedness. They shall not strike their feet so as to reveal details of their hidden ornaments. You shall repent to God all you believers, so that you may succeed." 24:31
khimar mean a cover, not a headcover. It could be a dress, blanket, anything. What normally shows? It could be the hands, face, hair, (lower) arms, (lower) legs, etc. Yes, there are some that have corrupted the meaning, but the Quran does not say headcover, that is clear.