Having participated in several MCI drills, The biggest initial problems are lack of communications and manpower. Even at WDW if they hit the panic button, it is still going to take at least 15-30 minutes before you can get enough manpower to address the throngs of fleeing, panicked and injured people. During the initial stage of the event, confusion, miscommunication, and misinformation are prevalent. Drilling helps, but even the best law enforcement agencies are susceptible to this. I don't think you're going to with OCSD and surrounding agencies responding to Disney not being trained and ready. But I do think that they will be very overwhelmed at any major incident that happens at WDW. I would imagine that DS is probably their biggest concern for the fact that it is not a controlled space. Unlike the theme parks which I would consider hardened targets, DS is open, accessible, packed with people, and it would be identified with Disney World. Which would pack the same psychological punch.
Side note, I know cast members get a background check. I would assume that there is no screening for them before they enter the parks?