If we divide everything into rational/irrational, the OP sounds correct. If, however, we look at it in terms of the mind (logic) and the heart (emotion), we get a different picture.
????? ???? ????? ?? ????; emunah and bitachon start with the mind, and our job is to have our hearts join our minds.
I think if someone has enough bitachon, it probably would overcome anxiety. So I disagree about the two being entirely unrelated. However, most, if not all, people don’t actually have a high enough level of bitachon to overcome severe anxiety.
Thetefore, I don’t think that the way to approach it practically is to work on bitachon (although, of course, that’s a good idea for everyone, regardless of anxiety), I think it should be dealt with as the professionals would recommend (perhaps CBT?). The reason one person suffers from anxiety more than another is not usually because he has less bitachon, it’s because he has more anxiety. So I agree about the da ma shetashiv.