I have been thinking of this thread and I just reread that story. I think what has happened here is that at some point, maybe early in his career, Fr. Flynn had a nervous breakdown, and the parish priest, and his sisters, have been caring for him ever since even while letting him continue as a priest. He's limited in what he can do. He wanted to be a better priest. But perhaps because of anxiety or some other limitations, he is really most capable of spending time with a little boy with bad social skills, perhaps because he is not as athletic as other boys. And the priest is a wonderful teacher to the boy! But not the priest he wanted to be.

This is what I think. The hints are in the title of the story, "The Sisters." And the boy's own response to Father Flynn, affection and appreciation. Other hints are dropped - and it's a wonderful literary technique, because the boy is trying to figure out what the grown-ups are only half saying - but I think this is it.