Isnt marriage and oneness the primary reason God created Eve and gave her to the silently-suffering Adam?
Marriage is a command by God: Therefore shall a man leave his father and mother, and cleave unto his wife - and they shall become one flesh.
I would say you are close to having that one nailed. And I will discuss what I am about to pose to you but I will not debate this one more time, discuss only.
This biblically based truth disturbs, by far, most of the people in the world today because, by and large, almost all of us are guilty of ignoring or learning the truth too late. As you have mentioned, God/Jesus brought Eve to Adam and the coupled, nice word, huh, and they were one. If we go forward we should notice that Father Abraham, when he was Abram, was married to Sarai but they is no ceremony noted.
A little further and I believe Abraham arraigned the Marriage of his son, Isaac. When Abe's servant returned with Isaac's Bride, she went into the tent of Isaac's mother and then Isaac went into her, they were married. And when God gives us a model we should pay attention but when god says it twice He has, so as to speak, underlined it.
But, and that is a large but, if God says something or God gives us a model three times, we really had better pay attention and God records Jacob/Israel working seven years for the wife he had picked out but her sister, Leah, was placed into Jacob's bed on his wedding night and after they had coupled, they were married.
There lies the issue, Scriptural Wedding not some custom man dreamed up and placed a tax on but God's view of the institution He created for the good of all mankind. And some foolish man invented the Drive In Movie Theater and another the Back Seat in the Automobile. All those divorces and they did not even realize they sinned.