All the times I've come off my bike, I held on to the handlebars for dear life as the bike went down, hit the ground, seperated & rolled. It's not like there was ever time for much else.
If you have time to plan how to catch yourself, you had time to avoid the crash to begin with. Just going with whatever the bad situation threw at you is often better than trying to keep your old plans or exercising control as the situation unfolds.
This works for high-speed snowboarding as well.
FWIW: 1 concussion, 2 broken helmets, & no broken bones to speak of.