That seems to be more than a little splitting hairs. There’s at least one other example of Jesus referring to a woman as “woman”, certainly the fact that her name wasn’t mentioned doesn’t mean the she didn’t have a name.
Also, how is God working through “a common, sinful teenager” created by God any more astounding than Him working through a common, sinful set of parents to see Mary born? Throughout the Bible God intentionally uses the weak, those we would not expect, from Jacob to David to Peter, so that we would not glorify those through which He works, but that all glory would go to Him.
Mary is to be greatly admired for her decision to submit to and pursue God’s will, even when that will presented her with a very troubling situation, but the credit goes to God for everything she was able to do, so I see no reason to lift up one person over another simply because God has chosen one for a great task and another for a humble task. We ought to honor people not according the the greatness of the task set before them, for that is God’s decision and not theirs, but rather by their faithfulness and obedience to that task.