Dovv: If all we’re talking about is fear of public speaking, then don’t worry about it.
But when it is more than that, (ex. anxiety about life, marriage, jobs, careers, children, avodas hashem, learning, etc.) then I don’t think bitachon should be the main focus of dealing with it.
I am aware that many rebbeim, especially in the yeshivos you have attended (based on things you have said), speak only about mussar, and never about emotional health, but there are others who say that emotional health comes before that. Including the Rosh Yeshiva. (RH)