So the story begins with Abraham. He was old - I mean, he lived a long time (longer than anyone these days does), but this happened when he was already a century old. His wife, Sarah, had a baby. And you know what? She was old, too - she was 90. Well past child bearing years. Yet suddenly, here was this senior citizen having a baby. That’s where the story starts. You see, Sarah gave birth to Isaac. Abraham had been promised, and he was looking for the fulfilment of this promise, that he would have descendants as numerous as the stars in the sky - or as the sand on the seashore. You know what, though? Isaac wasn’t the answer to that promise; he wasn’t the one who had so many children. He was, technically, something of a non-entity. No nation came to name itself after Isaac. He was mainly remembered in a formulaic way - because he came in between two “really” imporant people (as if he himself weren’t). You see, Isaac’s son (Jacob) became the father of the “large nation,” the one who “produced” the descendants as numerous as the stars in the sky, for Abraham. And God renamed Jacob Israel - and his descendants were known by his name.
And so we see that when God’s people (Abraham and Sarah) were really looking for something specific, God didn’t give it to them - He gave them the medium through which they would receive what they were looking for, and He did so by a miraculous birth. Sarah, in her old age, gave birth to Isaac - and he fathered Jacob, the start of Israel.
Consider Elkanah. He had two wives - Hannah and Peninnah. He loved Hannah, but she couldn’t bear him any children. She desperately wanted to have a son, and God did grant this to her - though she didn’t hold him to herself, but offered him up to the service of the Lord, under Eli the priest. This was, of course, Samuel. Now, at the same time as this was happening, the nation of Israel was wanting something. They wanted a king, like the other nations. All through the book of Judges, we are reminded that “there was no king in Israel, and every man did as he pleased.” It is clear that Israel needed someone to keep them in line (for some reason, they wouldn’t let God Himself keep them in line). If we follow the same formula as above, when God’s people are desperate for a specific something (a king, in this case), and there is a miraculous birth (certainly, Hannah was only able to bear Samuel because God specially gifted this to her), then the miracle child will not be the thing that the people desire - but will be the one who leads to the thing desired.
As Isaac was gifted to Abraham and Sarah, and he eventually fathered Jacob (whose sons became known as a nation, called by his own name given to him by God, Israel), so Samuel was gifted to Hannah - and even more, to Israel - and he eventually gave them the king that they wanted so much. First Samuel anointed Saul for this work, but he also anointed David, when the time came - and David has always been recognized as the finest king that Israel ever had. So while Samuel wasn’t the one who fulfilled the wishes of the people in himself, he was the one who brought about the fulfilment of their wishes.
Again, then, we see a miraculous birth. Elizabeth is past the age of child bearing. Yet she is gifted by God to bear a son. At this time, God’s people are looking for a Messiah - someone to save them, deliver them, from their enemy. John, again, is not the one to do this. John is the one who will point ahead to the One that will. John is the child who is born of human parents (just like Isaac was; just like Samuel was), by divine intervention, for a specific purpose. He points to the One that God’s people are looking for; the One they need. He points to Jesus.
Now, there’s something about Mary’s pregnancy that sets it apart from these other miraculous pregnancies. Mary is a virgin. Jesus is not the product of God “opening” a womb that is otherwise beyond child-bearing years. Jesus is not the product of God giving special dispensation to a couple (Elkanah and Hannah) who have otherwise been unable to have children. Jesus is something totally new to the Biblical narrative. Never before had God done something like this; never again will He. Jesus was pointed to by John, but John was a forerunner (in a similar sense to Isaac being the forerunner of Jacob, or to Samuel being the forerunner of David). Jesus pointed to Himself - He is the Way, the Truth, the Life; He is the Vine; He is the Bread of Life.