TL;DR: Yes, there is damning evidence that the relationship between the Catholic Church and the abuse of children has been systemic.
According to a recent report by the UN (CRC/C/VAT/CO/2), as reported by CBC:
The Vatican "systematically" adopted policies that allowed priests to rape and molest tens of thousands of children over decades, a UN human rights committee said
In its report, the committee blasted the "code of silence" that has long been used to keep victims quiet, saying the Holy See had "systematically placed preservation of the reputation of the church and the alleged offender over the protection of child victims."
The report itself states:
29. The Committee is
particularly concerned that in dealing with allegations of child sexual abuse, the Holy See
has consistently placed the preservation of the reputation of the Church and the protection
of the perpetrators above children’s best interests, as observed by several national
commissions of inquiry.
43. ... Well-known child sexual abusers have been transferred from parish to parish
or to other countries in an attempt to cover-up such crimes, a practice documented by numerous national commissions of inquiry. ... Due to a code of silence imposed on all members of the clergy under penalty
of excommunication, cases of child sexual abuse have hardly ever been reported to the law
enforcement authorities in the countries where such crimes occurred. On the contrary, cases
of nuns and priests ostracized, demoted and fired for not having respected the obligation of
silence have been reported to the Committee as well as cases of priests who have been congratulated for refusing to denounce child abusers, as shown in the letter addressed by
Cardinal Castrillon Hojos to Bishop Pierre Pican in 2001;
60. The Committee expresses serious concern that in dealing with child victims of
different forms of abuse, the Holy See has systematically placed preservation of the
reputation of the Church and the alleged offender over the protection of child victims. The
Committee is particularly concerned that while the Holy See recognized in its written
responses and during the interactive dialogue the primary competence of judicial
authorities, it has continued to address these cases through Canon Law proceedings which
contain no provision for the protection, support, rehabilitation and compensation of child
victims. The Committee is also particularly concerned that:
(a) Child victims and their families have often been blamed by religious
authorities, discredited and discouraged from pursuing their complaints and in some
instances humiliated, as noted especially by the Grand Jury in Westchester, the Ryan
Commission in Ireland and the Winter Commission in Canada;
(b) Confidentiality has been imposed on child victims and their families as a
precondition of financial compensation; and
(c) Although it has extended its own statute of limitations, the Holy See has in
some instances obstructed efforts in certain countries to extend the statute of limitation for
child sexual abuse.
So the answer according to the United Nations Committee on the Rights of the Child is yes. There is a systemic relationship between Catholic priests and sexual abuse of children in that the church has:
adopted policies that allowed rape and molestation of children over decades;
a systematic preference for preserving the reputation of the church and sexual offenders above the best interest of those the church has abused;
demoted or fired those who violated the "cone of silence" on child abuse, and congratulated those who have refused to denounce child abusers;
obstructed efforts in some countries to extend the statute of limitations for child sexual abuse;
blamed, humiliated, discredited or discouraged its victims of sexual abuse.
I noted that the report cites the Winter Commission (Canada) regarding the Mount Cashel affair I mentioned in the question.