It's not just Cof E who have these problems. I was raised Catholic and 60 years ago the RC church had a custom of "pastoral visits" from the parish's priests (all male of course) to the parish's RC homes to give pastoral/spiritual/practical advice/care.In principle a good thing one would have thought.
One such priest visited our family home, on a rota presumably , he was new and rumoured to be much "rotated" around the diocese's parishes.
I was only 10+ years old and I do remember him but he struck me as a pleasant, mild-mannered man and I took no particular notice of him.
Many years later when I and my two sisters were all adults , my late beloved mother found out to her fury that he was rotated because he was a danger to children/young people. We were never alone with him and one didn't anyway know his preferences so girls might have been uninteresting to him. I know nothing about any problems he presented nor what he may or may not have done in the wider parish, if anything.
He may have kept his hands clean, been a reformed character by the time he reached us.
Someone with relevant experience once made to me the very obvious point that these characters who want to abuse children will always try to get near to them hence their prevalence in children's homes , schools etc.
This stuff is everywhere in these sorts of institutions. Unfortunately.