Trust is so easily lost and so difficult to regain.
Words rarely work to regain trust. Only changes in behavior over time are helpful.
I think the steps you mentioned to regain trust are right. It takes time. You can't force it.
Your family want to trust you. They are rooting for your recovery. But they also need to protect themselves against disappointments.
But I think that it isn't really trust you need as much as some empathy for how difficult the struggle is.