I'll let Asteriktos keep going, because I enjoyed his post. Just a couple of points short on bibliography.
IMO, Mary plays such a small role in the Gospels (in terms of how much she factors as a character in the overall narrative) because the Gospels are not about her. They, like she, are all about Jesus. The gospel is the "gospel of Jesus Christ the Son of God" (Mk 1.1). St John concludes his Gospel by saying that only certain things, things necessary for our salvation, were written from among all Jesus said and did, and were they all to be written down, the whole world could not hold the books necessary for telling the story (cf. Jn 20.30-31, 21.25). Implicit in that claim, to me, is the idea that, if anything more needed to be written, the priority would always be Jesus. It is St John who records our Lady's last known words in the Gospels: "Do whatever he tells you" (Jn 2.5). So even her priority is her Son, leading others to him and fading into the background, one among the crowd, but also uniquely "the One", the "blessed among women".
I forget which AFR podcast it was, but I remember hearing Fr Hopko talk about our Lady as a "mystery of the Church". The idea was that the fundamental preaching of the Church, her main message, concerns Christ. "We preach Christ crucified" (I Cor 1.23), as St Paul taught. When someone comes to believe in Christ, comes to accept Christ, comes to repentance and baptism, is made one with the Body of Christ through Communion, when one irrevocably "puts on Christ", only in that context will talking about Mary make any sense. Only within the Church can she be understood as she is. Outside of that context, she'll always be misunderstood, and we see that with the Protestants and others outside the communion of the Church.
Christ compared himself to the grain of wheat that falls into the earth and dies in order to bear much fruit. In a way, Mary is the pearl that we do not cast before swine, because swine are incapable of appreciating it. The pearl is kept for the heirs of the kingdom, because above all others royalty appreciate jewels. Pearls adorn the children of kings, and she is the crown and adornment of a humanity redeemed by the King of kings, she is the pride of our race, she who gave birth to Emmanuel.