Why did Joseph Smith write The Book of Mormon in Early Modern English (thee, thou)?
To acquire something of the authority associated with the Christian Bible of his time.
It seems to me that any answer to this question is likely to offend at least one group of people, so let me say from the outset that this is my opinion as a non-Mormon.
As I said to a couple of young Mormon evangelists, as I returned a copy of the Book of Mormon to them:
"It reminds me of the old joke: Your work is both good and original. Unfortunately, the parts which are good are not original and the parts which are original are not good."
To me, the Book of Mormon seems to be extensively copied from the King James or similar Bible, with Joseph Smith's own additions, which attempt to copy the style and language of the rest of the text.
Joseph Smith was writing into a culture that to some extent revered the Christian Bible, which used the Elizabethan English language.
So by using a similar style and adding it to the Bible, his book acquired something of the authority of the Bible for those who encountered it.
As Irenaeus of Lyons said in the preface to "Against Heresies":
"Error, indeed is never set forth in its naked deformity, lest, being thus exposed, it should at once be detected. But it is craftily decked out in an attractive dress, so as, by its outward form, to make it appear to the inexperienced (ridiculous as the expression may seem) more true than truth itself."