I saw the movie. It definitely presents George as important for winning public support for the war, which is nonsense.
It's important that the King be able to carry out his duties. George and Elizabeth were very significant figures in the British propaganda campaign; that's a matter of historical record. And the movie hardly polemical about anything; it takes the monarchy as an important British institution, which it was at the time of the movie, and still is today. It critiques some of the isolation and snobbery involved, but portrays the development of cross-class understanding.
This personal story is in fact interesting. But it would have been just as interesting if George had just needed to master public speaking for a job, say, teaching. Or, at least, it should have been. I don't think this movie really believes that which is why all the tripe about the importance for humanity of the King's speech.
But George wasn't a school teacher. He was the King of the Commonwealth & Empire.
And the stakes are much higher and much more public, which the movie acknowledges. The same way the stakes of FDR being crippled by polio are much higher than with someone who isn't a head of state.