I've only had a couple of serious falls, and in both cases, it was wholly a matter of instinct and physics. I didn't have time to let go of the handlebars or take any kind of evasive action whatsoever. I was a ballistic particle.
I had a nasty break of my pinky finger playing basketball years ago, and the result was a full hand cast. I was going nuts not being able to ride my bike, until I talked the doc into letting me ride with aerobars. As long as I stayed on the flats and took it easy, I could brake well enough to stop.
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Proud parent of a happy inner child ...