What is the significance of the pattern of miraculous births in the Bible?
One of the first things to recognize is the infant mortality rate during this period of time. The article "Children in the ancient Middle East were valued and vulnerable — not unlike children today" under the heading "The death of children" mentions:
Archaeological data shows that the infant mortality rate was 50 per cent in the ancient world, but children who died were often cared for.
So it is not surprising that the Biblical record highlights certain "miraculous" births.
The next point we see in the examples from the question is the significance of some of these births. Note the particular births of the OP points # 1, 2, 3, 5, and 6. These are all births that are connected to the Messiah. What does this tell us? The Jehovah God was carrying out his will for the Messiah to be born and given witness to. The fact that God's name means "He Causes to Become" as the appendix of the New World Translation brings out:
However, this definition well fits Jehovah’s role as the Creator of all things and the Fulfiller of his purpose. He not only caused the physical universe and intelligent beings to exist, but as events unfold, he continues to cause his will and purpose to be realized.
As to Hannah and the birth of Samuel, this is just another example of Jehovah establishing his will for his people. Samuel was one of the great prophets of that time. But let us not forget about Samson's mother that was also barren (Jg 13:2, 3) and Samson becomes one of the notable judges of Israel. A Shunammite woman (2Ki 4:14-17) was also barren; while not much is mentioned about the child, the woman showed great faith in Jehovah's prophet Elisha. The Biblical account attributes the birth of Obed (Ru 4:13), an ancestor of the Messiah, to Jehovah God.
As the "source of life" (Ps 36:9), Jehovah God has shown throughout the Bible that he can and will make sure that his purpose to "fill the earth" (Ge 1:28) is accomplished.
[All scripture quotations from the New World Translation of the Holy Scriptures (Study Edition)]