How important is the historical verification of Biblical events to traditional Christian faith?

How would traditional Christian theology suffer if there were little or no verification of the Bible's historical content?

As there have been ~2000 years since the bible took place, this means there has been extensive amount of time that has gone into verifying much of its history based on location/names, making this question opinion based. (As OP references Book of Mormon its also important to note that the Book of Mormon does not state explicitly where it takes place, making verification difficult)

That said, Christian theology would not suffer very much. All you have to do is examine the parts of the bible that haven't been verified. The most obvious takes place in the garden of Eden. Genesis 2:10-14

10 Now a river went out of Eden to water the garden, and from there it parted and became four riverheads.

11 The name of the first is Pishon; it is the one which skirts the whole land of Havilah, where there is gold.

12 And the gold of that land is good. Bdellium and the onyx stone are there.

13 The name of the second river is Gihon; it is the one which goes around the whole land of Cush.

14 The name of the third river is Hiddekel; it is the one which goes toward the east of Assyria. The fourth river is the Euphrates.

Even though this place can't be verified it does not detract from the message the bible contains or that other stories from the Bible can be verified.

Even when places can be verified (Jerusalem, Bethleham, road to Damascus, etc) often exact locations aren't specified. Just knowing that a place exists does not verify that an event took place (or that it didn't). Quoting comment to OP:

I don't believe Paul simply because Ephesus is a known place any more than I believe in the goddess Artemis because Ephesus is a known place. I trust Paul for other reasons. I see supporting arguments from history, but foundational arguments from the witness of the Holy Spirit (see 1 Cor 2:4)

4 And my speech and my preaching were not with persuasive words of human wisdom, but in demonstration of the Spirit and of power,

Or in Jesus' words after his resurrection (John 20:24-29)

24 Now Thomas, called the Twin, one of the twelve, was not with them when Jesus came.

25 The other disciples therefore said to him, “We have seen the Lord.” So he said to them, “Unless I see in His hands the print of the nails, and put my finger into the print of the nails, and put my hand into His side, I will not believe.”

26 And after eight days His disciples were again inside, and Thomas with them. Jesus came, the doors being shut, and stood in the midst, and said, “Peace to you!"

27 Then He said to Thomas, “Reach your finger here, and look at My hands; and reach your hand here, and put it into My side. Do not be unbelieving, but believing.”

28 And Thomas answered and said to Him, “My Lord and my God!”

29 Jesus said to him, “Thomas, because you have seen Me, you have believed. Blessed are those who have not seen and yet have believed.”

I believe verification is unnecessary to Christianity when we can read the words, listen to the spirit, and know the truth. Evidence is nice to have (and there is evidence) but it should/is not be the basis of theology.

Before attempting an answer to your question I needed to clarify what is meant by Christian theology in the light of historical content. The word history comes from the Ancient Greek ἱστορία (historía), meaning inquiry, knowledge from inquiry, or judge. History is the study of past events, particularly in human affairs. So what does this have to do with theology?

The word “theology” comes from two Greek words that mean “the study of God.” We study God’s Word to discover what He has revealed about Himself. He is the almighty, self-existing, self-determining Being with a mind and a will—the creator God who has revealed Himself to humanity through His Word, and through His Son, Jesus Christ. It seems to me that what you are asking about is Biblical Theology. Here is an extract from an article on that subject:

Biblical theology is the study of the doctrines of the Bible, arranged according to their chronology and historical background. In contrast to systematic theology, which categorizes doctrine according to specific topics, biblical theology shows the unfolding of God’s revelation as it progressed through history. Biblical theology may seek to isolate and express the theological teachings of a specific portion of Scripture, such as the theology of the Pentateuch (first five books of the Old Testament) or the theology contained within John’s writings, etc. Or it may focus on a particular period of time, such as the theology of the unified kingdom years.

Many credit J. P. Gabler, a German biblical scholar, with beginning the field of biblical theology. As he was being inaugurated to a professorship in 1787, Gabler called for a sharp distinction between dogmatic (systematic or doctrinal) theology and biblical theology. For Gabler, biblical theology must be strictly a historical study of what was believed and taught in the various periods of biblical history, independent of modern denominational, doctrinal, philosophical, or cultural considerations. In general, the principles that Gabler espoused were correct, and he influenced the development of biblical theology for many years to come. Biblical theology traces God’s progressive revelation on a specific topic and helps us see the Bible as a unified whole, rather than as a collection of unrelated doctrinal points.

Source: https://www.gotquestions.org/biblical-theology.html

If that is the basis for your question, then I must say that the verification of historical events as described in the Bible is very important. Otherwise, if there was a complete lack of evidence to support the places, peoples and events of the past as described in the Bible, we would have no intellectual basis for placing our faith in God’s word. We might as well be reading about Narnia.

As Jason Alls has already observed in his answer, the prophecies in the Bible show us the power and foresight and truth of God. “They help humans to see that God is who He is.” And that must be our starting point – God’s perspective – because if God has lied to us and the Bible is simply a fairy story, then there can be no sustainable theology.

God has not revealed everything to us and, indeed, why should He? The purpose of God’s word is to reveal to humanity how God deals with His creation and to point us to the ultimate solution to sin, which came with His Son, Christ Jesus. By looking back in time we find evidence to support the peoples, cities and events as described in God’s word. We may not find answers to every question posed about the historical events in the Bible, but we can see how nothing is hidden – even the imperfections and sometimes downright wickedness of the characters involved are laid bare. We get an honest account, one that exposes the faults and failings of the characters described.

A huge amount of verification exists to support the biblical account of events, peoples and places as described in the Bible. Some people fail to grasp that the Bible is alive, it is powerful, it moves people. It’s not just some good read or a source of comfort to people who are afraid (although it’s that, too). It changes lives and transforms people.

We study God’s word to discover what He has revealed about Himself through His word, and through His Son, Jesus Christ. It is through such study that we discover the revelation of God’s qualities and truthfulness and we come to know who He is. That’s what theology is about. The historical verification of places, peoples and events merely supports the veracity and truthfulness of the written word. What if there were little or no verification of the Bible's historical content? But there is. If there were not then the millions of people who have placed their faith and trust in God’s word would never have come to saving faith in the Bible and in Christ Jesus, the Son of God. Biblical theology traces God’s progressive revelation on a specific topic and helps us see the Bible as a unified whole, rather than as a collection of unrelated doctrinal points.

Personally, I don’t need some historian to verify that the towns of Sodom and Gomorrah existed. I don’t need some scientist to give credence to the order in which creation happened (notice I do not say evolved). All I need to know is that God has done what He said He would do and that I can trust his word – both written and in the flesh. His word is alive, and powerful, and transforms lives.

How important is the historical verification of Biblical events to traditional Christian faith?

It is important that the Christian faith is not effectively contradicted by historical evidence. The Christian faith is a faith based on historical events.

If any of the events can be shown not to have happened then the faith collapses.

Why did Paul say "If Christ be not risen from the dead your faith is vain and you are still in your sins"? (1 Corinthians 15:12-21) He said it because our faith is dependent on the historical event of the resurrection.

But obviously the Christian faith is not dependent on historical events for every individual believer.

A little child comes to faith in Jesus because they believe they are a sinner and need a Saviour just like him.. nothing less will give them peace.

Most come to faith in Christ before they have much clue of the historical evidence in support of the faith.

The Christian message speaks most powerfully to the conscience of the sinner and presents itself as suitable to the needs of sinners:

God made the heavens and the earth for his own glory; God made man to seek him, find him, and love and worship him; the whole human race is fallen in sin and rebellion against God; God is holy and must and will punish all sin on the Great Day that is coming; if you die without being forgiven then you remain unforgiven and shall be punished for your sins for all eternity; there is nothing you can do to make good your fallen sinful state except call upon Christ the Sinless One who took the punishment of sinners no better than you to have mercy on you turning back to God in repentance and faith; so give your life wholly over to God as his child, servant, and trust in the full forgiveness of every sin for Jesus's sake; do not delay for you do not know how much time you have left in this world.

This message has a loud, self-authenticating ring about it, even if most sinners suppress the self-authentication, holding down the truth in unrighteousness (Romans 1:18).

I love to find extra-biblical historical and scientific authentication of the Bible. It is my main hobby. But for hundreds of years before the beginning of the 1900s the amount of verification from science and archaeology was very limited.. yet there was the golden age of the puritans and huge evangelical revivals under John Wesley, George Whitfield, etc. Later, in the 1900s about an extra 10% of Scotland was converted very rapidly in a major evangelical revival. I strongly doubt you will find any of the messages of the preachers of those revival times focus at all on extra-biblical verification, either historical or scientific.

Very important. This is evident in the Bible itself from God's perspective and not man's. Let me explain.

God is very strong in the scriptures about his honesty and trustworthiness. Even if God would rather not do something, He has to do it if He previously said He would for the sake of His Good Name. Because of His Good Name and Character, God cannot and will not go back on anything He has said, ever!

With this in mind, I refer you to the prophecies from God through out the Bible from Genesis to Revelation. These prophecies seek to show humans the power and foresight and truth of God. They help humans to see that God is who He is.

If we look at people like Nostradamus and Baba Vanga, they do not prophecy the words of the Holy Spirit. Their sources of prophecy are from non-Godly sources. This is evident in that not all of their prophecies are true. Yet, all of God's prophecies do come true as our expert witness, world history, testifies to.

For example, the ancient Hebrew version of the geneology in Genesis 5 provides this prophecy:

Man given mortal sorrow. The Blessed God shall come down, discipling. His death shall bring the despairing comfort, rest.

ADAM = MAN

SETH = APPOINTED, GRANTED, GIVEN

ENOSH = MORTAL, FRAIL, FEEBLE

KENAN = SORROW, DIRGE, LAMENTATION

MAHALALEL = THE BLESSED GOD, THE PRAISEWORTHY GOD

JARED = SHALL COME DOWN

ENOCH = DISCIPLING

METHUSELAH = HIS DEATH SHALL BRING

LAMECH = THE DESPAIRING

NOAH = COMFORT, REST

So, we can see that God has given us foresight through this geneology, that a feeble man of mortal frame will come who is greatly afflicted and of God that will die, and that his death will bring the suffering comfort and rest.

In Isaiah 9:6, again we see that a human will be born to us. But this time it is more explicit, and states that this man is God our Father, the Holy Spirit who will come in human form:

Isaiah 9:6

For to us a child is born, to us a son is given, and the government will be on his shoulders. And he will be called Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace.

So, in Isaiah 9:6, God has validated who the person of Genesis 5 will be. Further validation that Isaiah 9:6 is speaking about the same person is mentioned in Isaiah 53:

Isaiah 53

53 Who has believed our message and to whom has the arm of the Lord been revealed? 2 He grew up before him like a tender shoot, and like a root out of dry ground. He had no beauty or majesty to attract us to him, nothing in his appearance that we should desire him.

3 He was despised and rejected by mankind, a man of suffering, and familiar with pain. Like one from whom people hide their faces he was despised, and we held him in low esteem.

4 Surely he took up our pain and bore our suffering, yet we considered him punished by God, stricken by him, and afflicted.

5 But he was pierced for our transgressions, he was crushed for our iniquities; the punishment that brought us peace was on him, and by his wounds we are healed.

6 We all, like sheep, have gone astray, each of us has turned to our own way; and the Lord has laid on him the iniquity of us all.

7 He was oppressed and afflicted, yet he did not open his mouth; he was led like a lamb to the slaughter, and as a sheep before its shearers is silent, so he did not open his mouth.

8 By oppression[a] and judgment he was taken away. Yet who of his generation protested? For he was cut off from the land of the living; for the transgression of my people he was punished.[b]

9 He was assigned a grave with the wicked, and with the rich in his death, though he had done no violence, nor was any deceit in his mouth.

10 Yet it was the Lord’s will to crush him and cause him to suffer, and though the Lord makes[c] his life an offering for sin, he will see his offspring and prolong his days, and the will of the Lord will prosper in his hand.

11 After he has suffered, he will see the light of life[d] and be satisfied[e]; by his knowledge[f] my righteous servant will justify many, and he will bear their iniquities.

12 Therefore I will give him a portion among the great,[g] and he will divide the spoils with the strong,[h] because he poured out his life unto death, and was numbered with the transgressors. For he bore the sin of many, and made intercession for the transgressors.

Now, bear in mind that it is God who is the author and originator of our Christian Faith. And God validates what He says and shows throughout the scripture. He does so, so that generation afer generation can know they can trust His Holy Written Word.

God further validates this person of Genesis 5, Isaiah 9:6, and Isaiah 53 in Zechariah 9:9. We read:

Zechariah 9:9

The Coming of Zion’s King 9 Rejoice greatly, Daughter Zion! Shout, Daughter Jerusalem! See, your king comes to you, righteous and victorious, lowly and riding on a donkey, on a colt, the foal of a donkey.

We can acknowledge that this person is the person Jesus of the New Testament, as in Luke 19:28-40, we read:

Luke 19:28-40

Jesus Comes to Jerusalem as King 28 After Jesus had said this, he went on ahead, going up to Jerusalem. 29 As he approached Bethphage and Bethany at the hill called the Mount of Olives, he sent two of his disciples, saying to them, 30 “Go to the village ahead of you, and as you enter it, you will find a colt tied there, which no one has ever ridden. Untie it and bring it here. 31 If anyone asks you, ‘Why are you untying it?’ say, ‘The Lord needs it.’”

32 Those who were sent ahead went and found it just as he had told them. 33 As they were untying the colt, its owners asked them, “Why are you untying the colt?”

34 They replied, “The Lord needs it.”

35 They brought it to Jesus, threw their cloaks on the colt and put Jesus on it. 36 As he went along, people spread their cloaks on the road.

37 When he came near the place where the road goes down the Mount of Olives, the whole crowd of disciples began joyfully to praise God in loud voices for all the miracles they had seen:

38 “Blessed is the king who comes in the name of the Lord!”[a]

“Peace in heaven and glory in the highest!”

39 Some of the Pharisees in the crowd said to Jesus, “Teacher, rebuke your disciples!”

40 “I tell you,” he replied, “if they keep quiet, the stones will cry out.”

I could continue this narrative of God's historical verification of events by His prophecying the years in advance, and having people write them down when they happen for future generations like ours to read, validate, and accept.

The people of Luke's time had the Old Testament prophecies to identify the coming Messiah. Today, we have the New Testament that testifies to and verifies the Old Testament narrative. We have the Old Testament that testifies to and verifies the New Testament. And we have the many wonderful examples of God's loving kindness towards the sinful human race in the many testimonies of people who have been converted, exorcised (delivered), and healed even to the present day.

Two very good examples of historical Bible event verification that led to the conversion of two very staunch athiests are James Warner Wallace, author of Cold Case Christianity and Lee Strobel the author of The Case for Christ. Both these people became Christians on the weight of evidence that verifies the truth and validity and historical accuracy of the Old and New Testament writings upon which we as Christians base our faith.

And so our faith in God and the scriptures is not a blind faith but an evidencial one that has been time and battle tested to stand the tests of time. And we as Christians can walk confidently in the knowledge that accepting the free gift of salvation through Jesus Christ and repenting of our sins will lead us to eternal life with God our Eternal and loving Father.

Peace be with you.