The Catholic priesthood has likely always been disproportionately same-sex oriented. St Peter Damien complained about a "sodomite" Church-within-the-Church in the C11th. Emmanuel Le Roy Ladurie's classic study Montaillou found evidence from the C14th of homosexual networks which were urban and clerical. Since Church teaching gave them no legitimate social space or erotic life other than celibacy, of course same-sex oriented men went into the priesthood: it gave their celibacy "more value" and was a way of "proving" that God did not hate them, of being "the best little boy in the world". It was well understood for generations that that "that sort of boy" would be steered into the Church.
What happened in the postwar period was that the social space for homosexuality expanded, so that those who went into the priesthood to escape their sexuality were paying an ever higher price in what they were giving up. It is not surprising that abuse increased. (I am referring here not to paedophilia as such, but primarily the inappropriate sexual relations with teenage boys.)
But, and here's the big "but", the same general patterns of social change were operating for all the Churches. It is the Catholic Church specifically which had an abuse problem of particular intensity and extent, both because it had so many priests who were refugees from their own sexuality and because the hierarchy—as celibates themselves—were far too solicitous of erring priests and far too dismissive of the pain and suffering of the abused. Being celibates, they did not have the visceral "it might be MY child" response one would get in the Protestant churches. Being priests, they held protecting priestly authority to be the most important issue, and, following Catholic teaching on sex, they judged sex in terms of its form, not in terms of its human experience. It is all very well to talk about "the Sexual Revolution of the 60s" but one needs to also focus on the very specifically Catholic nature of the extent of the problem.
Note also that I am talking of the internal procedures of the Church, not their interaction with the civil authorities. Laying down Church policy for a global institution means covering a vast array of different polities and regimes, with the Church having often have very fraught interactions with various regimes. The Nazis, for example, used accusations of paedophile priests in an orchestrated attack on the Church in response to Pope Pius XI’s attack on Nazism. As a knowledgeable interlocutor has said in an email to me:
… it's impossible to understand the Church's "cover-up" policy, as we'd now call it, without understanding its relationship to civil governments going back to the Investiture controversy, Garibaldi, the Masonic French Third Republic, etc etc. The Nazis were just one instance (though one much in mind since the Nazis were still in Rome in 1944.)A “one size fits all” policy has inherent problems. A city of Boston that is democratically run, has a Catholic majority—particularly among judges and priests—and operates in a democratic federal Republic where overt Christianity is entirely respectable is not quite the same as dealing with Nazi Germany. Indeed, one can argue that, precisely because the civil authorities were so friendly (and known to be so) that the Church hierarchy was particularly inadequate in its response within the Boston archdiocese. (Something similar may also apply in Ireland.)
But, the issue of making policy for the globe and such a range in Church-state interactions conceded, the internal failures of the Church loom large. Made worse by senior Vatican figures giving every appearance of being lost in their own moral universe, as the Pope’s preacher compares attacks on the Church over child abuse to anti-Semitism, the Vatican Secretary of State claims paedophilia is linked to homosexuality (not true and the Vatican is now trying to back away from) while Vatican radio blames an “anti-Catholic hate campaign”. They come across as morally tone deaf in a profound way, that just makes everything worse.
Some of the criticism is unfair: but mostly the Church has made a rod for its own back and is continuing to do so.