The study of the New Testament is an enormous discipline that has had more thought and academic labor put into it than any other field of science. The fact that Lee Strobel's book is written for a popular audience means that the reader will not fully appreciate the enormous amount scholarship that lies behind the many quick conclusions he makes. He is ultra-condensing. That is the strength and weakness of the book; a strength in that it makes the otherwise imposing New Testament scholarship accessible to laymen, and a weakness because the person demanding more will be disappointed. The book should serve as a launching pad for further discussion and delving deeper into his conclusions. That is why Strobel has a list of suggested reading after each chapter.
One of your complaints is that he does not interview anyone from the opposing side. I agree with you that this is another weakness of the book. It is obvious that any book would be far greater if it fully presented the other point of view and then showed why that view was wrong, and "The Case for Christ" is no exception. Then again, it is "The Case for Christ" and not "The Case against Christ". The scholars that he interviews do in fact (albeit briefly) deal with the major arguments against the trustworthiness of the New Testament. Whether you like it or not, the scholars he interviewed are in fact the experts, and are qualified to make the statements that they do. Another problem with interviewing the opposing side is the problem of choosing which opposing view to showcase, because opposition scholarship is not in agreement in its quarrel with the New Testament. I think Strobel did the right thing in choosing the format he did, and if I were to seek out expert advice on why we can trust the New Testament, I would not have gone anywhere else.
Now let us come to the most important thing. Your point, that using the Bible to prove the Bible is circular reasoning, is flawed. You are failing to appreciate that the Bible is the source-book on the life of Jesus and the early Church, and it must be recognized as the true source-book that it is. This is not circular at all because we are not talking about whether the Bible is inspired Scripture or not. Now if we were to say, 'the Bible is true because the Bible says it is' we would have a problem. However, that is not what we are doing. We are examining the Bible just as we would any other historical book, and are evaluating its claims for what they are. We then arrive at the conclusion that it is trustworthy and inspired only after the study of it. To say that one cannot turn to the Bible in order to examine the trustworthiness of its claims is nonsense.
You are absolutely right that one of the major arguments for the trustworthiness of the New Testament is that due to the early date of its composition, if the information in them was false the whistle would have been blown on it. This is a powerful argument, not easily dismissed. I believe you are failing to see its power (your objections show that failure). You say that Christians think it takes a long time for legends to form. That is not the argument at all. Legendary tales can obviously form immediately after an event, and that is easily proved as you have shown (Smith and Washington). But this is the actual argument: a controversial legend cannot form in close proximity to its reported occurrence without some form of refutation if the event was public and not private by nature. Many of the most important events in Christ's life cannot be legendary because the reports of them were given in close proximity to their occurrence and did not receive refutation though they were public and not private by nature. For example, the Gospels claim Jesus rode into Jerusalem on a donkey accompanied by crowds of people waving palm branches and singing his praises. It reports that the religious leadership was disturbed by this and confronted Jesus, commanding him to put a stop to it. It reports that this occurred just a week before his very public trial and crucifixion. It reports that his death was instigated by the religious leadership in Jerusalem and permitted by Pontius Pilate. These events were publicly proclaimed and interpreted by the apostles in Jerusalem not long after Jesus's crucifixion and alleged resurrection. The point is: if these were false stories, the opponents of Christianity could easily have refuted them, and due to the controversial nature of Christianity, they would have had perfect incentive to do so. However, what these reports did receive from the opponents of Christianity was reinterpretation, never refutation: that is, the earliest opponents of Christianity never refuted the reported events of Christ's life, but reinterpreted them, arguing that Jesus was a false teacher, sorcerer, was crucified as an object of God's wrath, and that his disciples stole his body from the tomb on the third day. Make no mistake: the Jews did not want Christianity to succeed. They actually boasted that the Pharisees did the right thing in condemning Jesus to death. But never did an opponent say, "That never happened! I was there!", though they would have had ample opportunity and incentive to do so. Rather, they repeatedly gave a different interpretation of the events than the disciples gave. The fact that the Gospels record all sorts of controversial public events in Jesus' life within a short time of their purported occurrence is powerful evidence for their truthfulness in light of the absence of refutation from the opposition, and the presence of reinterpretation by the opposition. That is the argument.
Regarding your second point about eyewitnesses, all of the Gospels were written within a relatively short time of the life of Jesus. As you have said, even many liberals date the Gospels within a early window (50-100 AD), while conservative scholarship dates them even earlier. Some of the letters of the New Testament were written even earlier than the Gospels and contain all the relevant information about Jesus contained in the Gospels, thus confirming the veracity of Gospels. Many studies have shown that an internal examination of the Gospels reveal them to be eyewitness accounts. While certainly drawing from a common tradition, they also bear enough discrepancy to show that they are actually independent of one another (adding even more credibility). The unanimous testimony of the early Church writings confirm that they were eyewitness accounts or were based upon eyewitness accounts. In short, we have many solid reasons to believe they were eyewitness accounts and no reason to believe that they weren't. Hostility to the Gospels stems not from the facts, but from a disinclination to believe the Gospels claims.
Your question about the Mormon eyewitnesses is a good one. I answer by saying that there is no comparison with the Gospel eyewitnesses and the Mormon eyewitnesses. Their claims were not public but private, and an examination of their lives lead many to believe that they were liars seeking gain. By contrast, the claims of the eyewitnesses of Christ were public, and an examination of their lives lead even the toughest critics to believe that they were sincere. They gave their lives for their faith in Christ, and taught the highest moral truth known to earth. Anyone who reads the New Testament with a thoughtful heart cannot but be struck by this.