Regarding your first point, of course the Bible can be trusted as an "historical" document in the sense that it was in fact written at some point in history. But that's not the question. The question is whether or not it is a sufficiently reliable witness to history--that is, does it document actual history accurately.
Your chart includes references to other historical documents. But you would never insist that these other documents are perfectly actuate historical chronicles--that is, that the events described by Homer actually happened exactly as he described them. Rather, you would recognize that Homer had his own biases that distort the "actual" history and that, in some cases, he took great poetic license. Any fair reader must read the Bible in exactly the same way--as being an imperfect witness to history that contains biases, poetic license, severe editing and censoring over the centuries, etc.
I will specifically address your other points in a book that I'm publishing about the Bible here on Steemit. I hope to post the second chapter later today, though it won't be till I get to Part II of the book that I really begin to address your points in detail.
Infallibility is the output conclusion, not the input assumption.
The purpose of these videos is not to make any case for infallibility. They only make the case that we know with very high confidence exactly what the original authors were claiming.
That takes away the intervening 2000 years and lets us confront the Bible as a collection of eyewitness accounts whose credibility we must assess.
After reading those accounts, many people simply find the authors believable. Specifically, they conclude it likely that God really did take on a physical body and supernaturally prove beyond their own initial doubts that He was who He said He was.
It is only after you decide to believe those witnesses that you face the problem of how to treat an accurate rendition of the testimony they left behind. If you have already accepted that Jesus was capable of supernatural acts and that he instructed his disciples to pass on His teachings to their successors, then it is a relatively tiny leap to believe their claims that they had supernatural oversight in writing their documents.
If you somehow believe that Jesus rose from the dead, yet do not believe that He preserved an accurate account of His teachings, then you are still faced with the question of which of these two defense strategies you want to rely on at your Final Performance Review.
Hmmmm. Your call.
So, in practice believers wind up assuming infallibility because there is no one alive with the authority and credibility to countermand a teaching in a way that we trust to stand up in the Lord’s court.