It does irreparable damage to the sanctity of the church. The appeal of Catholicism is supposed to be "I'm a bedrock of faith, I'm unchanging, I'm incorruptible." Kings themselves feared the institution, to the point where one was excommunicated and spent days walking barefoot around his castle to show penance.

One stain on that white cloth strips it of divine authority, and they're not interested in having that anyway, it would seem. A pope could theoretically create an entirely new wave of Catholicism by simply executing the perverts in front of the Papal throne.