It's an interesting question, whether there is such a thing. On one hand, Freud himself was a lay analyst. On the other hand, there's probably a reasonably large contingent of (orthodox) analysts who would assert that it can't exist for structural reasons (the setting, the lack of payment, the nature of the relationship, etc.).
If we had a solid reductionist biological theory of mental illness, we would still sometimes discover that the cause was strictly "outside" the patient's own physiology.
Psychologists certainly go along with the fashionable talk of "chemical imbalances", but unlike psychiatrists, they don't really claim any authority when they do it.
My general impression is that the only way to communicate how psychiatry is would be to write a very thick novel with characters who occupy many different positions in relation to the profession. Even then it would be hard to give an all-encompassing impression.
There is such a thing as "lay psychoanalysis", not involving a professional. I wouldn't dismiss it.
What I would definitely dismiss, as someone who has spent time in a psychiatric hospital as a patient, is the bad faith pseudoscience of psychiatry. Individual doctors may be doing their best, but the profession is very much as described in this article.