Polarbear, those are legitimate points! I often wonder about handedness myself and try switching hands occasionally, partly out of curiosity and partly for a challenge. I originally fenced left, for instance, but right felt more natural.
I tend to lead my lower body on the right side (e.g. flutter kick swimming freestyle, running, kicking) and I have a slight right-dominant strength advantage weightlifting, despite my best efforts to the contrary. Maybe I'm better described as mixed handed with right tendencies?
My mother has the same tendencies.
DS showed strong lefty proclivities for a while and now seems ambidextrous. I try to demonstrate new activities with both sides to avoid the bias you mention.