Consider this gif from wikipedia. All the orbits in the animation have the same orbital period $T$ and the same semi-major axis $a$, but different semi-minor axes $b_1,b_2...b_5$. This shows that the orbital period is independent of the semi-minor axis.
As an analogy, imagine twin figure skaters spinning with the same angular momentum. The red figure skater holds a weight halfway away from her body for a full revolution. The pink figure skater holds the same weight far away for part of a revolution, but close in for the other part of the same revolution, in a way that the average distance of the weight from her body is halfway. Both figure skaters will have the same period of revolution, but the length of the path of the weight will be larger for the red figure skater!
The time for a revolution for each skater is a function of the average distance they hold out the weight, rather than the difference between the closest and furthest distance they hold the weight!
Similarly, given fixed masses, the period of an orbit is ONLY dependent on the mean distance between the bodies: $a$. It is NOT dependent on any variables relating only to variations in distance between the bodies (i.e. eccentricity, semi-minor axis).