>you just need to know at least a little more than the companies you are consulting for.
sometimes (i'd argue often, actually), you don't even need that. simply having an outside/fresh perspective and the fact that you aren't part of any of the existing groups/silos is valuable.
Probably true, but to get the job in the first place you probably need some sort of showy, impressive credentials.
Often the most useful thing is just listening to the right people in the company. I wouldn't be 100% surprised if someone in the company in the story had already had the idea for the third electrode, but it took the suggestion from the high-paid consultant to get it taken seriously.