Before going into the detail of the answer I'd like to turn your attention to the correct translation of " بِمَا فَضَّلَ اللّهُ بَعْضَهُمْ عَلَى بَعْضٍ" which is:
because of the advantage Allah has granted some of them over
There are a few points by which your question can be answered:
1. Society will be established even if as few as two people get together.
2. Society is in urgent need of someone to be in charge who is capable of managing affairs, skillful enough to make the final decisions in grave situations and qualified to guide the society.
3. Theoretically speaking, both man and woman can possibly be the manager of this small society which is called "Family" but they can not be in charge at the same time as too many cooks spoil the broth.
4. As you have mentioned in your question, Allah has appointed "man" as the manager of the family. There are some reasons why "man" has to be the manager in a family:
Men possess much greater judicious prudence which means that they are stronger, braver and more capable of undertaking tough tasks which requires forbearance and courage while women's character is dominated by feelings and emotions which means that they are good at taking up special tasks which requires delicateness. Leadership (not dictatorship) is one of the tasks requiring courage, bravery and strength especially when the family plunge into some chaos. We cannot expect to feel comfortable by lying and putting our head on a piece of rock, likewise, it doesn't make any sense to use pillows in order to build a bridge. Pillows are useless in building bridge as rock is of no use for lying and taking rest. So when building a bridge we can say that rock has an advantage over the pillow without devaluing and ignoring the importance of pillow.
In Islamic point of view, No Rights Are Granted To Anyone Unless Some Obligations Are Imposed To Them. man is obligated to provide for his wife. He has no right to compel his wife to carry out the household chores. In return he has been granted some rights: 1. His wife has to submit herself to his sexual desire and cannot prevent him from having sexual intercourse with her. 2. His wife cannot go out of the house without his permission. (Source)
5.The rights mentioned above have nothing to do with Men and Women as human beings. When it comes to the humanity the only criteria is wariness of God:
يا أَيُّهَا النَّاسُ إِنَّا خَلَقْناكُمْ مِنْ ذَكَرٍ وَ أُنْثى وَ جَعَلْناكُمْ شُعُوباً وَ قَبائِلَ لِتَعارَفُوا إِنَّ أَكْرَمَكُمْ
عِنْدَ اللَّهِ أَتْقاكُمْ إِنَّ اللَّهَ عَليمٌ خَبيرٌ
O mankind! Indeed We created you from a male and a female, and made
you nations and tribes that you may identify yourselves with one
another. Indeed the noblest of you in the sight of Allah is the most
God wary among you. Indeed Allah is all-knowing, all-aware.(49:13)