Asking this question since the first password manager
Audits are but a means to try champion that trust. And, indeed by no means a silver bullet at that.
Yes but, depending on how the ecosystem is built, the amount of trust needed can be smaller or greater. Reality isn't black or white, we also have shades and colors.
They have some fascinating papers about it, if I recall.
What has happened to some password managers though is that they don't store the metadata encrypted (like username, website name, etc.) so that leaks have revealed which sites you use but I don't think any decent password manager has leaked passwords without a client being hacked, right?