Part of the reason I intrinsically trust start ups over BigCos is because they do not have such teams. Instead, I would expect the start up to be more personal, hands on, response and human. AirBNB, as much as I love them, seems to be falling flat on its face in this regard.
I tend to have the same view, yeah. I think there are situations where less bureaucracy in that regard can actually produce a more paralyzed response, though, depending on the personalities of all involved. For example, if you have a legal team but not a crisis-management team, the founders can be receiving far too cautious advice, and a not-legally-knowledgeable founder might not feel confident ignoring it. A good crisis-management advisor would instead give advice that balances legal risks with some sort of assessment of which of the legal risks are worth taking, while the legal department is more likely to tell you to avoid anything risky.
