No. Correct analogy.
While there are modes of failure for an automatic transmission, there are modes of failure for cambio corsa, friction, index and electronic shifting. There can only be no mode of failure with fixed gear. As I wrote before, Henri Desgrange was correct.
There are fewer modes of failure with fixed gear, but there are still failure modes. Your chain can still break, your tires still puncture, your BB/hubs can have damaged bearings etc. A poorly set up fixed gear will have as many problems as a poorly set up derailleur system, and a properly set up derailleur system will have as few problems as a properly set up fixie, but with the added benefit of being able to go uphill with 1 wheel turn/pedal rev and down the other side with 4 wt/rev.