I know in a poetic and/or religious sense, we hear people speak; of the good Angel and the bad Angel having their influence on us.

I don't believe this...

But I wonder, what causes the conflict between the two hemispheres of the brain.
I have read that great geniuses including some great artists;
are able to think with the whole brain in balance.
That their great works and insights are achieved during this brain equilibrium.
And that at times, we are all able to do this.

There are programs out there, such as Brain Entrainment, that are supposed to have the effect of Whole Brain Thinking.

I have some of these Sound Therapy programs; and I find them fascinating.
But it's all momentary...
You have to continue using the programs to get a momentary 'flow'.  And even if you achieve more then a momentary balance-like-flow; you still have to continue with the programs to keep it.
What I wonder is this:

Why, in all the Wisdom of the Body and Mind's workings, would it choose to make this separation of the brain's hemispheres?  Which causes conflict in the body-mind-self.
And could there ever be a time and a way, to 'think' with the Whole Brain in Balance?

Not just for short or even extended periods of time...

A permanent balance.