I know a therapist who does trauma therapy, and he said that he doesn't expect his clients to ever fully trust him. He said he still is going to try to be in the room with them, be as safe as he can be, and try to help while they walk through the pain of processing and integrating trauma and healing.
Part of healing from trauma, for some people, is about learning to have boundaries about what they are ready to do and not ready to do.
Trauma taught me it was dangerous to let go. For me, trust means letting go. My therapist told me it's ok if I don't trust her or even if I get angry at her sometimes. She believes it is a part of the process of healing, and we are constantly working on what helps me to feel ok enough to keep doing the work and stay in therapy.
There are many good reasons to tell our therapists as much as we can. For some people, the work they need to do in therapy is to disclose more and take more risks in that way.
In my case, taking responsibility for my own healing sometimes means I need to slow down my disclosure. My therapist deliberately slows me down, just so that I don't re-trigger myself by trying to tell her more than I can face telling her. This is one example of how taking responsibility and trusting a therapist sometimes can look very different than disclosing everything we can. In my case, it means actually slowing down and listening to myself on what I feel like I am ready to share and not ready to work on yet. Trusting in therapy means trying new ideas, and considering new things... including slowing down and making my own decisions about what topics I want to work on and not work on.