“King James did have a very questionable personal life. but he was not directly involved in doing the translating.”
Indeed. In retrospect, I wish they had named the translation something along the lines of “The Royal English Bible” or “The English Reformation Bible” instead. That way, it would take some focus off who happened to be the monarch at the time.
The Puritans and the Nonconformists did not fully embrace the KJV until after the Restoration.
The KJV is really "The Anglican Bible" since it was translated to be appealing to proto-High, Low and Broad Church factions in the C of E.
Andrewes (High), Rainolds (Low) and Barlow (Broad) were a very representative cross-section.
Officially, it is the “Authorized Version.”
The “King James” title evidently arouse informally and was adopted by convention; I don’t think it is considered an official title.