To be clear, This was clearer at the start but we have veered into a more narrow discussion. I am not saying the only thing that matters is who says yes first. The real problem is the entire rigidity of our recent dating system. The halacha that a guy must give the yes first is but one example, but try not to get too caught up on that.
Here is a useful rule of thumb: (for life too) When a complex problem is deemed to have an easy-quick-fix-solution, the simplistic solution will rarly solve the problem