There are two aspects to such scandals: that the abuse occurred in the first place and the reaction of the Church hierarchy to cases of abuse.
Patterns of abuse
The former seems to be a mixture of opportunity (priests and other religious were placed in regular and authoritative proximity to vulnerable children and teenagers) and patterns of motivation. If one felt one’s own sexuality was problematic, then the celibacy of becoming a priest (or other religious) became a refuge: a place to hide from one’s sexuality. This includes genuinely problematic sexuality (notably paedophilia) and sexuality made problematic only by doctrine (homosexuality). Hence, according to American data (which is the most comprehensive) the only group where a majority of victims were female were the victims aged 7 years or less and almost a third of the victims were aged 15-17, with the proportion of male victims rising with age.
It is clear that the Catholic priesthood has always been disproportionately single-sex attracted. Peter Damien complained about a “sodomite” “Church within the Church” in the C11th; Ladurie’s classic study Montaillou found homosexual networks in the C14th which were urban and clerical; while modern estimates of the proportion of Catholic priests who are homosexual range quite highly, but it is clearly much higher than in the general population.
Given that the same-sex attracted are a persistent proportion of the population and Catholic doctrine essentially requires them to be celibate, of course they took refuge in the priesthood: it made their celibacy useful, provided status and “proved” that God did not hate them. It was a way of being “the best little boy in the world”.
Which, as I have previously discussed, was probably a relatively stable situation until gay communities started coalescing after the Second World War. As the legitimate social space for being homosexual expanded, those choosing the priesthood as a refuge were, in effect, paying an ever higher cost in foregone opportunities. Hence the pattern of rising rates of abuse from the 1950s to the early 1980s followed by a sharp drop. The drop being a product of the priesthood stopped being a “safe haven” for abuse as priests were increasingly prosecuted; training for the priesthood improved; and more same-sex attracted priests were motivated by genuine vocations than “making the best of bad options”.
The moral failure of the Church hierarchy
The reaction of the Church hierarchy had three features: complicity, denial, acceptance that there was a genuine problem, with each feature being dominant in turn.
Priestly celibacy not only made the priesthood a refuge from personal sexual turmoil, it also meant that the official Church response was by male celibates. That is, men who were not fathers so who lacked the visceral reaction of “that could be MY child” which clearly marked the reaction of many Catholic parents once they became aware of the extent of the problem.
Lacking that visceral reaction, it is clear that the hierarchy’s dominant concern throughout has been protecting priestly authority. Hardly surprising, it is the dominating motivator of the behaviour of the Catholic Church generally. For example, most of those who organised the Holocaust were born and raised Catholic yet no one was ever excommunicated for the Holocaust. Conversely, if you challenge priestly authority, then excommunication is extremely likely.
So—in order to protect priestly authority—for decades the Church hierarchy was essentially complicit in abuse: shuffling abusing priests around rather than dealing with the problem, trying to deny the extent of the problem and only publicly dealing with it when it had got to the stage that failing to do so would cause even more damage.
But there was a more subtle problem with Church doctrine. Catholic (and Orthodox) teaching on sexuality is the Christian natural law position: the final cause of sex is procreation so the only legitimate form of sex is unimpeded penile-genital intercourse within marriage. Despite the burbling on about the joys of “unitive sex”, human experience is irrelevant: the form is everything.
So, for example, if a husband gives his wife a massage, that is fine. If he also gives his wife orgasms from oral sex, that is evil. The pleasure of the wife, that it be an act of love by the husband, that is all irrelevant: such sex is wrong in form, so evil.
Which means the human experience of sex does not matter, only the form matters. (Remembering it does not matter if the husband is sterile and/or the wife post-menopausal--i.e. procreation is impossible--as long as the form of unimpeded penile-vaginal sex within marriage is adhered to.) Now, as it happens, Jesus had something to say about thinking form matters over human purposes, but that apparently does not count.
But, if it is the form of sex that matters—not the human experience of it—then that encourages adherents of such views to not merely discount the human experience of sex in joy (as they clearly do: the anathematisation of homosexuality profoundly discounts human experience and human agency) but also in abuse (as they clearly did in so many child abuse cases).
So, Catholic doctrine and practise is directly connected to the moral failures—particularly the massive moral failures of the Church hierarchy—exposed by the abuse scandals.
While it is tempting to quote Matthew 12:33-36 on knowing things by their fruits, the more appropriate passage is when Jesus talks about what is or is not “unclean” and quotes Isaiah:
"'These people honor me with their lips,
but their hearts are far from me.
They worship me in vain;
their teachings are but rules taught by men.' You have let go of the commands of God and are holding on to the traditions of men."