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These symbolic coincidences could also be considered as proof, or at least a strong indication, of the truth of the Bible.
Suppose someone publishes an article that contains the line "Jerked Conifers will be seen in the Cyder Pines." (assume that it makes sense in the context it appears).
Two years from now, when "Frederick Jones" assumes the "Presidency", people will notice the anagrams and realize that that was actually a concealed prediction.
If that article contains several other hidden predictions that come true, there will be little doubt that there was something very special about it. (And who knows how many other predictions it contains that haven't happened yet.)
This same effect happens with types in the Bible. A story or event seems ordinary and self-contained, but then much later, something happens that matches the original description in ways that are unlikely to be coincidence. When enough of these happen, coincidence becomes almost impossible.
For instance, at the time they were instituted in the Torah, no one realized that God's seven holy feasts were actually symbolizing his plan of salvation. It's only in hindsight, knowing what was later explained in the Greek scriptures, that their meaning becomes clear.
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Why is [typology] in the Bible?
This question seems to veer close to opinion/speculation, as it's basically asking "Why does God...?"
Nevertheless, God created us as intelligent creatures in an intelligible creation, and reveals Himself intelligibly.
The Sacred Authors themselves make use of types, e.g. Matthew 2:15
in fulfilment of the word which the Lord spoke by his prophet, I called my son out of Egypt.
Compare that to Hosea 11:1
Soon fades the dawn; soon passes king of Israel. Israel in his boyhood, what love I bore him! Away from Egypt I beckoned him, henceforth my son.
St Matthew uses Israel as a type of Christ in the flight to and return from Egypt.
Another example is Hebrews 1:5:
Did God ever say to one of the angels, Thou art my Son, I have begotten thee this day? And again, He shall find in me a Father, and I in him a Son?
Compared to 2 Samuel 7:14:
he shall find in me a father, and I in him a son. If he plays me false, be sure I will punish him; ever for man the rod, ever for Adam’s sons the plagues of mortality;
So what is it for? As Anthony Maas, who wrote Christ in Type and Prophecy puts it, "[Typology] furnishes solid and reliable premises for theological conclusions" when the typological sense is either revealed as such, or when it is proven from Scripture or tradition. 1
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Not all analogies between OT and NT character / event / story qualify to be interpreted as typological relationship. We should be guided by how Jesus and NT authors themselves use them, or probably only use typology that NT used. Or we should only use what the Apostolic Fathers or the Early Church Fathers like St. Irenaeus or St. Augustine used. Otherwise, we can potentially go overboard with our imagination and creativity such as several examples you provided in the OP.
As to the purpose, I think it's quite established in many theological circles that the purpose is mainly to show fulfillment of OT prophecies by Christ, progressive revelation, and thirdly to compare the many aspects of our pilgrimage on earth (under Christ's Lordship) in terms of Israel's journey to the promised land (under Yahweh).
This Got Questions article shows Biblical use of typology while this Wikipedia article shows a few other common ones.