Why might a very small tension crack have a strong influence on factor of safety under saturated conditions?

Section 3.1.4 of the Swedge Theory manual describes the difference in assumed pore pressure distribution for a model with and without a tension crack. 

Basically, if the model has a tension crack, the peak pore pressure is at the base of the tension crack.  If the model does not have a tension crack, then the peak is at the midpoint of the line of intersection of the two joint planes.  It’s similar to the distribution assumptions made on pages 5 and 6 of the “A Slope Stability Problem in Hong Kong” chapter of Hoek’s Practical Rock Engineering notes for planar failure.  It is this difference in assumed pore pressure distribution which results in the differences you are observing.  Also note that having a very shallow tension crack will result in a Factor of Safety close to the dry condition, since the maximum pore pressure is relative to the depth of the tension crack (equation in section of the Swedge theory manual for a model with a tension crack).