You also pay for the psychological friction and pressure that the consultant has to suffer from the fact that customer expects them to be geniuses with magical powers and the project managers planning the time line according to that.
I know one guy (a consultant now) who I would trust to Navy SEALs parachute into almost any technical problem and come up with a real solution. Any part of the stack, from building a physical hardware test bench, debugging firmware compiler bugs, desktop software development, to a scalable web stack. These people exist, and they're not cheap.
However, it's worth noting that no one with skills like that works at Deloitte.
It makes me think of "The Wolf" from Pulp Fiction.
"It's about thirty minutes away. I'll be there in ten."