You sort of hint at this, but one thing these organizations seem never to take into account is the staff time required to review what could amount to hundreds of entries, if not thousands. This will require developing some sort of system to review all the entries (time), the time to review them, the time and energy wasted in being paralyzed by too many choices. (Despite evidence to the contrary, we still think more options=better.)
One could say that working with one firm for months would take the same amount of time or longer, but that time would be valuable time spent shaping the entire brand story of the organization. Time is not equal.
What usually drives this is, I believe, an underlying lack of confidence in being able to adequately vet a good designer. And spinning the story that engaging the community is a good idea. Both are valid issues. But contests are not the way to do that.
Especially since a contest means that other people are defining your brand instead of you defining it.