Well, having watched your video, I have a few concerns.
First, the discussion of Old Testament textual criticism is phenomenally misleading, as it addresses only Masoretic principles of textual preservation, which date to up to a millennium after the actual composition of the texts they are preserving. By the time the Masoretes got the texts, they had already been circulating and getting edited and redacted for literally centuries. Rabbi Richman's claim that there are no discrepancies between the most ancient manuscripts and the ones we use today is a laughable falsehood. I was responsible for putting together the critical apparatus of several chapters of Isaiah for the new BHQ edition, and I found textual variants in literally every single verse of every chapter for which I was responsible. The Septuagint, the Dead Sea Scrolls, and even the Vulgate and Samaritan Pentateuch show numerous, numerous variations in the manuscripts. While many of them do not affect the sense of the passage, many of them do, and quite significantly, too. This video presents either a grotesquely ignorant or a deliberately misleading picture of the nature of textual criticism and the state of the Old Testament manuscripts.
The claim that there has been no "systematic change or major omission" is also misleading. The Dead Sea Scrolls and the Septuagint show that, for instance, the Book of Jeremiah was a full 1/8 shorter in the versions circulating aroudn the turn of the era. The version you read in the MT has been significantly, and systematically, expanded. 1 Samuel is also vastly different in the Septuagint and the Dead Sea Scrolls. We might also point out that the Dead Sea Scrolls and the Samaritan Pentateuch show the Samaritans were actually right that the original version of the exodus tradition has the Israelites establishing the altar on Mount Gerizim and not Mount Ebal. Then there's the systematic use of the word "angel" to alter the meaning of passages where God himself was said to have visited and interacted with humanity. The Septuagint and Dead Sea Scrolls show that as well.
Then we have the very conservative Simon Gathercole appealing directly to faith by implying that Jesus' appeal to Isaiah must mean it was accurate. How silly. That has absolutely nothing whatsoever to do with textual criticism. He mentions that Jesus reads out a scroll from Isaiah, but he neglects to mention that that scroll does not match the version of Isaiah preserved in the MT. Does your Bible change that text in the Old Testament to match the version read by Jesus Christ himself? I'm willing to bet it does not. Don't you trust Jesus?
Next, on the New Testament, Dr. Williams is not being totally forthcoming by saying the Bible of the Christians inside and outside the Roman Empire agreed. They simply did not. Early New Testament manuscripts these days are actually grouped according to broad typological similarities. We have the Western family that has a specific collection of variations, the Byzantine family that has another profile, and the Alexandrian, which has a third profile. There are some other minor ones, too, but these are the major ones. It should also be added that their canons were quite a bit different from each other for centuries after the life of Christ. The picture this video and its editing is painting is incredibly misleading and dishonest (or naïve).
Williams later states that there's no problem establishing the original text because "if one scribe makes an error and fifty scribes don't make that error, we know what to do." The problem is that you have no empirical way of unilaterally determining what is and isn't an error. There are numerous errors that were proliferated by fifty scribes and avoided by only one. What happens if we just assume the one is the error? Well, we end up with vastly different manuscripts with more errors in them than agreement, which is exactly the situation. This video's editors and the conservative scholars they are quoting are promoting fundamentalist misrepresentations of the state of New Testament textual criticism.
I will end with this statement from Dr. Williams:
When you take all the manuscripts that we have, there are no variations of any significance.
This is flatly and demonstrably false. There are numerous, numerous variations of great significance. I've discussed them for years and Catherine has ignored them for years.