Even if the claim was usually true on longer time scales, I doubt that pointing out an organisations mistakes and not entirely truthful statements usually increases the trust in them on the short time scales that might be most important here. Reforming organizations and rebuilding trust usually takes time.
Yes, it’s important that the public should trust authorities. Pushing authorities to be more truthful helps, not harms, that goal.
Even if the claim was usually true on longer time scales, I doubt that pointing out an organisations mistakes and not entirely truthful statements usually increases the trust in them on the short time scales that might be most important here. Reforming organizations and rebuilding trust usually takes time.