For Arturia it isn't so much that the church had been involved at all, merely that in both Grail Wars she had seen the position of Overseer had been abused. If the Grail had a mind with any sense it makes sense to her for it to simply pick a Servant it knew to be of virtue to mediate the conflict.

"I would have thought there would have been people willing to aid in the cleanup as it was, but no matter. Every world, I've found, is different. Strange though it may seem, your Grail War and mine being different makes sense. Most likely the three families simply chose a different system in yours." It was an easy thing to picture happening. There was no reason Arturia could think of that the families couldn't adjust the system just slightly when they created the ritual in the first place.