I have trained as a psychotherapist and can tell you that Freudian theory has has not been completely abandoned, more superseded or integrated into modern psychotherapeutic theory and technique. That said, it's important to understand the distinction between psychoanalysis which as you suggest is not very popular any more, scientific / 'empirical' psychology which aims to understand phenomena through experimental procedures, and modern psychotherapy which comes in many flavours, some with psychoanalytical roots, others not.

In some branches of psychotherapy (e.g. psychodynamic therapy), Freudian theory is still considered to have some useful and good ideas by many psychotherapists such as the unconscious (we are not the masters of our own house - some of our mental contents / processes are unavailable to us), loss (possibly unresolved 'stuck grief'), symptoms (behavior or physical distress related to a mental functional disturbance), and intra-psychic conflict (within our psyches there can be some sense of competing agencies). The big one is the Oedipus complex, but even this taken loosely still provides a developmental narrative that says some interesting stuff about our earliest attachments, even if you reject the contents of the theory. I think all this stuff is interesting and in some cases illuminating of client material.

Linking this to literary theory, modern therapists are often quite into the idea of 'narrative'. The client brings you a story and we disrupt their version of events with our own therapeutic one - psychotherapy is a form of narrative criticism, if you like. As for the theories themselves, if you read Freud's case studies and papers, they themselves read like short stories and this is where things get interesting / weird; they can be taken as a type of literature in of themselves, and can therefore be legitimately subject to literary criticism and psychoanalysis in the same way they subjected his patients to a form of narrative criticism in their analysis.

So, I suggest this: all psychological theories, even scientific ones, are a form of literary criticism, brought to bear upon, and explain, stories of mental suffering. Therefore, why wouldn't Freudian and Jungian themes are so appealing to people who are looking for an evocative way to make sense of stories and the characters who inhabit them, such as literary critics for example? These people aren't looking for scientific truth (if that's what you mean by 'thoughtful analysis), but neither are they all avoiding 'thoughtful analysis' altogether - where do you get this idea? Of course, there are some pretentious literary critics out there who namecheck Freud but I think your question deserves a question in response: what do you think thoughtful analysis looks like?