Oh, and just as an addendum - I don't really see how biblical marriage roles have much to do with who should ask who out on a date. I would classify myself as broadly in the complementarian camp, but I don't have a prbolem with the girl asking the guy out. You are DATING, you are not MARRIED. Most people will date people that don't become their spouse - if you have had more than one date in your life, you are in this camp. Whether or not the woman prompts the first date or not says nothing about the man's future capacity to lead a family, if it gets to that. You can't get a read on someone's capacity to fulfill their role in a relationship BEFORE YOU ARE EVEN IN A RELATIONSHIP.
But look, if you want to think of it in terms of headship, look at this way - approaching a guy and asking if he would like to go on a date still puts the power in his hands to to decide what to do. Let him decide if he even wants to go out. Let him decide where. Let him decide on every subsequent step in any prospective relationship. But prompting that first move is not emasculating him, or removing headship, or anything like that. It's simply bring clarity out of what is otherwise guesswork.
But look, if you want to think of it in terms of headship, look at this way - approaching a guy and asking if he would like to go on a date still puts the power in his hands to to decide what to do. Let him decide if he even wants to go out. Let him decide where. Let him decide on every subsequent step in any prospective relationship. But prompting that first move is not emasculating him, or removing headship, or anything like that. It's simply bring clarity out of what is otherwise guesswork.