During the 19th century and first half of the 20th, cross-draw holsters and carry were far more common then it is now, yet the writen records of the time and memory of some older members, reveal no outstanding problems with people unintentionally shooting either themselves or others while drawing a handgun.
Admitedly there are some issues with today's game players, but back then folks didn't play these games. As with all methods of carry, the way to prevent an unintended discharge is to keep one's finger off the trigger until the muzzle is pointed in a safe direction. When one followes this rule cross-draw is both safe and can be very fast. However if games are your thing, or when shooting on public ranges, strong-side draws are better and less likely to cause others getting upset.