Why do some men, in their public prayer, use King James English? In their minds do they think it shows reverence or holiness?
The King James Version was a popular translation for almost 400 years, so its language has stuck in people's minds as religious or biblical language. But at the time it was translated thee, thou, thy, and thine were merely various forms of "you." Those forms have fallen out of use.
In 1611 these words were used to address people you were familiar with. “Used in middle English and in early modern English into the 17th century as the appropriate form of address to an intimate friend or a person of lower social status than the speaker” [Webster’s Third New International Dictionary]. That is why Quakers used “thee” and “thou,” to show that they considered everyone to be brethren. As these words fell out of favor, they became the formal way of addressing people and then eventually the form of addressing royalty and God. So while its use is continued today by some to address God, the meaning attached to the words have vastly changed from its original meaning.
There is nothing wrong with using archaic English words in a prayer, but it is wrong to claim that it is required.
The use of "thee and thou" is fine with me (not that is matters what I think!), it's when they use other old English terms. In my younger days I remember hearing prayers that sounded like Shakespeare and wondered how anyone understood it!