When I did the poll in the road cycling forum it worked out to about 3/4 of the people had fallen, and the common thread was that they either forgot they were clipped in until it was too late, or they unclipped on one side or not the other. It's not a balance issue, and I don't think it's a matter of luck, I think it's just a matter of being very conscientious about remembering that you are clipped in until the unclipping becomes natural so that you don't panic.
Clips add a new small but deliberate
extra step to an action that one has done many times before. As umd says, one has to be deliberate at first to add this new step as a habit. I suspect that a fair number of peole add clips and expect that magic happens!