You're right - copies were all made by hand (the printing press was yet to be invented). But these guys took their job incredibly seriously. If mistakes were made they threw the whole attempted copy out. Comparisons of the Massoretic text with earlier Latin and Greek versions reveal careful copying and little deviation during the thousand years from 100 B.C. to 900 A.D. We have an incredible amount of copies still today that agree with each other 99%.
Belief in the Bible requires less faith than many believe. We don't have any originals but we have so many copies that agree with each other. And this was the smallest of religions - a mustard seed, as it were. Deviated from Judaism with 1 - Jesus Christ, then 12 apostles, then missionary work eventually caused such major issues for Rome that they had to adapt to it and become "Holy Roman" in order to hold onto power. In spite of Nero feeding Christians to the lions, Christians would not recant their devotion to Jesus Christ, and this mustard seed continued to grow until Constantine stopped Christian persecution and decided he would embrace the Christians. I believe Constantine was merely being a politician and only embraced Christianity as far as it was politically beneficial to him. But I can imagine after a couple hundred years of abuse the Christians were probably fine (to some degree) with Constantine's membership. In spite of Rome's corrupting of Christianity (indulgences, the pope's ultimate power in spiritual as well as temporal matters), true Christianity remained unblemished as evidenced by Martin Luther's attempt to go back to it. In spite of Catholic corruption of the religion, true Christianity was preserved in the Bible. This movement has grown and grown and grown. And the evidence for it, other than improbable history, is the Bible. What the Bible said conflicted with what the Roman Catholic church was teaching (thank you Martin Luther). I have indescribable respect for the Bible's preservation that yes, my faith is strengthened, but it's not a blind leap of faith. It is supported by incredible evidence.
The "as far as it is translated correctly" line in the articles of faith is not specific to any translation errors, but is rather a subtle (though still vague) way to disregard the parts of the Bible that conflict with LDS teachings. I've quoted Paul to various Mormons and I consistently get an eyeroll "Oh, Paul said lots of crazy things." Well, Paul makes up the majority of the New Testament. Do they want to disregard the majority of the New Testament as translation errors? Some things Paul says conflict with mormon teachings (saved by grace alone and NOT after all we can do (NOT of works) lest any man should boast). Shrug. Any serious study of the Bible a mormon will come up against stuff that doesn't sit well with mormonism. So many just put it on the shelf and figure they'll understand the truth of it someday. But the Bible's accurate. And, if the mormon believes Jesus Christ is who He says He is, they might want to take these issues back off the shelf and find out what is true and what is not. If you only want to use LDS sources (instead of objective sources) - per LDS instruction - then just use those. When mormon apologists attempt to address controversial issues (Adam-God, people on the moon, Kinderhook plates, David Whitmer outing Joseph Smith's false prophesy about their selling the BoM rights in Canada, Book of Abraham papyri, differing versions of the first vision, B.K. Roberts' Studies of the Book of Mormon, Polyandry and polygamy, William Law's Nauvoo Expositor destroyed by J. Smith because he was blowing the whistle on Smith's polygamy) then these apologists will sometimes admit to some of the less than pleasant facts about mormon history. I love mormon history, but luckily I didn't throw out the baby Jesus with the bath water. I love Christian history too. My study of mormon history discredited mormonism, whereas my study of Christianity surprisingly strengthened my testimony of it.
You may not have researched this stuff (when can one find the time between church callings throughout the week and of course the necessary hours on Jazzfanz), but I think we can at least agree on one thing: Jesus is a stud.