This all boils down to modern thought and skepticism. People aren’t raised to respect authority, and are therefore individualistic, and question external authority being imposed on them. Conspiracy theories are formed as a hypothesis to explain why a particular authority has the power/money/wealth/influence that it currently does.
In my opinion, there are certain establishments that we should be skeptical of. For example, in the past 50 years, we have seen an erosion of our civil liberties and freedoms. Meanwhile money and wealth has been redistributed and concentrated in the hands of a few elites. This causes me to be skeptical of our representatives in the US government.
However, there are certain establishments that are trustworthy and proven. Their testimony is reliable, and we need not be skeptical of them. In the case of the Catholic Church, God himself guarantees that the gates of hell will never prevail, and I trust Him. Because of this, when conspiracies against the Church come up, I give the Church the “benefit of the doubt”. I also usually follow up with some research into the claims.
Because the Church is such a visible organization and has a lot of history behind it, it is easy for skeptics and critics to formulate conspiracy theories of the Church based oh half-truths and “itching ears”. Since the Church claims to be endowed with divine authority, the only rational things to do are either fully embrace it as a gift from God, or to abhor it and fight it with all their will–there can be no middle ground.
Therefore, people who believe in these conspiracies are really just skeptical of the authority the Church claims it has. The best thing to do is show these people the scriptural, historical, and philosophical basis for this authority. Depending on the persons background and disposition, you could focus more one one of these aspects than the others. I tend to use more of a scriptural basis for my protestant friends, and a historical and logical approach for the non-religious people I know.
Hope this helps shed some light on it! It’s merely my own experience, but hopefully it has some value to it! God bless!