meaning - What does “Church’s darkest existential problem in an era of existential problems” mean?

It is in a sense both.

Existential means, broadly, “characteristic of or concerned with existence”; but since the 1940s phrases like existential question and existential problem and existential crisis have meant questions and problems and crises of the sort discussed by the philosophers called Existentialists.

The catchphrase often recited as the core of Existentialism is “Existence precedes essence”. That is, the Existentialists rejected the notion, which for centuries lay at the core of Western philosophy (and particularly of the Catholic Church), that “Essence precedes existence”—that man is defined by ‘humankindness’, by membership in the class ‘human’, and there is thus a quality of being, an ‘essence’, to which the individual is called upon to shape his life and actions, his existence-in-the-world. No, said the Existentialists: the individual is free, is-in-the-world from the outset, and spends existence in creating a unique personal essence through action.

The author is saying, then, that the pederasty scandal confronts the Church with a problem which goes to the very heart of its fundamental identity—one which cannot be resolved by theological casuistry, by contemplation of the Will of God as defined by the Fathers of the Church, but only in its concrete actions in-the-world.