You said that King James of England was a Catholic, and therefore not satisfied with Protestant Bibles of his time, and therefore wanted a new version which would satisfy the elites of his day.

From what I've read of his history, King James I of England/VI of Scotland was raised by Protestant clergymen and their followers.  HIs mother, Mary Queen of Scots,
was a devout Catholic, but he was separated from her very early, and she was imprisoned by the Protestant Queen Elizabeth of England.  He didn't see her again.

During his particular reign, as I read it, it was illegal to be a Catholic.  The English went back and forth on this matter, whether it was God-breathed to be Catholic or Protestant.  During James' reign, it was considered divine to be Protestant.  

Now this does not mean that James may not have had a secret hankering for Catholicism.  Perhaps he pined for his mother, who had been executed for treason against
Queen Elizabeth.  Perhaps he listened to courtiers who were Catholic.  And perhaps therefore he wanted to make a Bible which was more satisfactory to both his Catholic courtiers (secret ones) and his Protestant (open) ones.  So what you say is true in a way.  He was known to want peace between the warring clergymen.  

But he was officially Protestant.  Some of his descendants, however, were Catholic and one of them, Charles I, got his head chopped off for being that way.  The
Puritans under Oliver Cromwell beheaded the king.  His son had to escape to France, where all good members of the Stewart family found refuge from time to time,
but came back at the Restoration.  At this point I am fuzzy about who was Catholic and who was Protestant among the reigning monarchs of the reigning Stewart family.
The monarchy went back and forth between the Pope and the Bishop of Canterbury so fast that people couldn't keep up.  Many of our ancestors fled to Ireland and then
America to save their hides from either a Protestant or a Catholic king or Prime Minister such as Cromwell.  I have ancestors who fled to Ireland to escape Catholics,
and ancestors who fled to escape Protestants.  Some of my Pollock ancestors changed their names to Polk to conceal their identities from the religious police.  President James Polk was a relative whose ancestors changed their name to escape persecution, but I can't remember whether it was Catholic or Protestant persecution.

Finally, the Brits got fed up with this ping pong, and outlawed all future Catholic monarchs.  They brought a member of the Stewart family, a Protestant member, back to file England, but she would only take the crown if they gave one also to her German husband William.  So we got the Protestant couple, William and Mary, and after that we had no more Catholic monarchs in Britain.

But now, the Commonwealth of the United Kingdom is gearing up to allow the monarch to marry a Catholic--but their children must remain Protestant.  So it goes on.

We may assume the ping pong match will end when Jesus becomes King of us all.  Hallelujah!

I know that there are some passages in the KJV which are better translated by other versions, as you say.  My Bible study compares passages between the various
versions, and they sometimes are quite different.  On that you are right!