Whatever the analogy is suppose to be, eternally-gratfull is trying to claim that when James speaks of a dead faith, James is saying that a dead faith is non-existing when James is actually saying a dead faith is a faith without works. The faith exists is just does not have works.
My "analogy"
If you believe/have faith a plane will not get you from point A to point B, you will not do the work of getting on the plane.
Therefore you have a belief/faith that prevents you from doing the work of getting on the plane.
So your belief/faith does not have the work of getting on the plane, it is dead but not non-existing for its existence is what keeps you from doing the work of getting on the plane. So your dead faith/belief exists, but it will not allow you to do the work of boarding the plane.
Yet if dead means non-existing, then you would not have a belief/faith that would keep you from getting on the plane. So why would you not get on the plane if the belief/faith that the plane would not get you from point A to point B did not exist?
It's a highly convoluted analogy if anything.