If your wife agrees to say the traditional wedding vows of: "Will you obey him, and serve him, love, honor, and keep him in sickness and in health? And forsaking all others keep yourself only to him, so long as you both shall live?"
If your fiancee agrees to these wedding vows, your marriage will last. If she doesn't agree to those vows, your wedding won't last. It's simple stuff.
A public promise? you know you can get married in private with the same vows right? It has nothing to do with declaring it publicly.
Do you even know what a promise is? Why people commit to them?
I'm not too sure what you don't understand about head of household. The buck stops there. It doesn't mean the man is right all the time.
It means the woman has decided to marry a man she deems strong and decisive and that those characteristics give her trust in his decision making and that she will respect them because of those attributes.
It doesn't mean obey him like a dog. Every partnership has a leader whether the parties believe in it, believe it, or see it. Men are supposed to be the leaders of their households.
Even in your "partnership", one of you runs the show whether you know it or not.
Jesus told you not to make any promises