I actually work at a Non-Disney park, and that, quite frankly is my biggest concern at Disney. At the park I work at we take safety very seriously, and between the sheer amount of inspection that gets done to every ride every day by multiple people and departments, combined with a great number of safety checks that occur prior to dispatching a ride, the park I work at has a great safety record, one of the best in the business. (I'm not going to say the name to keep ambiguity)
But I truly feel unsafe at disney, I am always double and triple checking myself, and the people around me, and I feel that many of disney's practices are unsafe. The only reason that I would imagine people feel safer there is the aura of magic, and the fact that disney rides are MUCH more tame then say a Six Flags or Cedar Fair park.
That's not to bash disney, obviously I'm a total Disney Dork, or I wouldn't be posting here, and the experiance at the Disney parks are the best I've ever had, but from personal experiance, they could definitely improve their safety checks, especially just to keep up appearences. For the most part, amusement accidents are just that, accidents, that either couldn't be prevented, or were the fault of the rider. That's not to say that things don't happen, they do, and they can happen anywhere, including Disney, which absolutly never gets bashed when an incident occurs at another park, like Cedar Fair, Six Flags, or other regional theme park companies do.