I am terribly sorry this mental health professional treated you in a way that was dismissive. It sounds like the psychiatrist was only interested in prescribing medicine — not actually engaging you in therapy — and managed to insult you in the process. It also sounds like it took a tremendous amount of courage for you to bring yourself into the appointment, and I deeply admire your persistence and willingness to contact a new therapist. I highly recommend someone other than a psychiatrist so the emphasis will be on therapy, rather whether or not you are a candidate for medicine. This isn’t saying a consultation with a psychiatrist might be helpful downstream — there are plenty of magnificent, compassionate ones — but for now let’s put the emphasis on explaining your needs to someone who knows how to listen and isn’t necessarily looking to prescribe.
You asked about the purpose. The reason therapy can be very helpful in the hands of a skilled therapist is that often the safety and relationship in the therapy has a great deal to do with correcting the issue. You said it just right: “If I don’t have any problems, what do you think those bad coping skills come from?” The answer to that is often rooted in prior relationships, often in the family of origin. The relationship with the therapist combined with techniques, interventions, and the process of therapy often change the dynamics beneath the symptoms.
I hope you find someone who can honor you needs better, and help make the changes in your life you are looking for. The Find Help tab at the top of the page can help you find someone in your area.