The history behind The Book of Mormon translation is based on written records. Who writes the history and what they have to say has a strong impact on how the events are understood. It is assumed that history comes fully formed in a textbook or what was written by an author who did the studies. For the modern historian, no history exists unless it is written down and somehow explained. Some physical evidence can be used to corroborate or refute the written record, but only words explain human thoughts and experience. People can only write from their perspective, and sometimes they lie or remember incorrectly. The Urim and Thummim found with the gold plates has a lot of evidence, while the stone in a hat a few strong statements. What can be known about the translation of the Book of Mormon depends on who and what to believe.
Remember that no one other than Joseph Smith was allowed to see, save perhaps Oliver Cowdery, the gold plates or Urim and Thummim before the translation was finished. The Lord had warned that anyone who saw them before given explicit permission would be destroyed. A warning of destruction applied to Joseph Smith if he showed them to anyone. To protect against this he always had the items covered or placed in a safe place, unless in use. After the translation the Lord gave permission to show the holy items to a select few. When the Book of Mormon was published, it contained the testimony of Three Witnesses that an angel showed them the gold plates. It also contained the testimony of eight other witnesses that they handled the plates. All official copies of the Book of Mormon through to the most recent contain the witness signed declarations. None of them ever denied the printed testimonies. On the contrary, the Three Witnesses later testified independently they saw the gold plates, the Interpreters, and other items.
How the translation was accomplished is a complete mystery known only to God, Joseph Smith, and possibly Oliver Cowdery when he was once given permission by revelation to try. Any statements about what Joseph Smith did or saw to translate the gold plates come second hand. None of them claim to be direct quotes from the Prophet, although they sometimes say that is where they got the information. Almost all of the descriptions are of a rigid reading. Mistakes found in the original manuscript pages that have survived refute such exact renderings; except for proper names and places. Whenever Joseph Smith does explain the translation in his own words, it is in the vaguest of terms. He does make it clear that the translation was from the plates using the Urim and Thummim that came with them by the Power of God. This implies having both together was essential to the translation work. Either he is absolutely truthful in his descriptions or he deliberately left out the stone in a hat portion of the process.