I can only relate my personal experience. I no longer receive any mental health services. But I used to see a psychiatrist. And I tried seeing a few mental health therapists as well. As you mentioned, the psychiatrist I saw was only interested in prescribing & monitoring psych med's. Except for my initial appointment which ran longer, my subsequent appointments lasted... oh... maybe 10 minutes & I was out of there. Therapy appointments were typically an hour & that's where pretty-much any-&-all discussion of concerns took place.

I never saw a psychologist. Where I live, at least, psychologists don't seem to do much in the way of ongoing therapy. (Maybe some do & I just never happened to see one.) Most of the therapy services that are available seem to be offered by people who refer to themselves as mental health therapists. Some of them may have psychology degrees. But many of them seem to have social work degrees. At least that's been my experience. (Perhaps where you live, things are different?) So, anyway, based on my own experience, I would say the place to start would be with either a psychologist or a mental health therapist depending on who is available for you to see. Then, if it is determined psych med's should be considered, that would be the time to see a psychiatrist.