We can trust it to not carry out a coup, to not get involved in domestic affairs, and to not play geopolitics without consulting with the DOS first, but that's about the extent that trust goes to.
For the others, you seem to be confusing 'ignorance of' or 'sentiment for' with 'trust'.
Police departments are federal, state, county, and city.
Religion is highly fragmented.
I imagine trust in "religion" would change if it was broke out into it's representative governing bodies as opposed to one big bucket.
Baptists don't trust the Pope or any Vatican leaders.
Catholics don't trust the Mormon church leaders.
Jehovah's Witnesses don't trust Jewish leaders.
and so on...
"Trust" in religion is almost always going to be "your own" religion, and even then, we see church infighting and splits regularly.