No, I don’t think you have the wrong idea. I think it’s unfortunate that over the last few decades, the helping professions have become very separated with psychiatrists, psychologists and medical doctors dealing with their patients separately. That leaves the consumer with the responsibility for coordinating their own care.
I suggest you look for a practice that has an interdisciplinary team. That means that there is a psychiatrist working alongside the various talk-therapists (psychologists, social workers, mental health counselors). When you call a practice or clinic for information, be sure to ask how much the disciplines in fact collaborate.
As far as your experience with a therapist: Not every therapist is a good “fit” for every individual. In my own work, I encourage people to “shop” by interviewing several therapists before signing on to do therapeutic work. As one of my colleagues pointed out (with no disrespect intended), finding the right therapist is like finding the right pair of jeans. You try several on until you find the one that fits you. In your case, you need to be working with a talk therapist who is more active/directive and who has a working relationship with your prescriber.
Please don’t be discouraged. There are many, many fine therapists in the world. You just have to give yourself the time (and the self-respect) to do your homework and find the right one for you.
I wish you well.