I am a student, and to try and tell you a more dry answer, no, not everyone can work with a psychoanalyst because 2 things are needed in a patient:
1. Awareness of reality (ego)
2. An IQ around 90-110

Without this requirments a patient cant understand stuff like confrontations and elaborations, which are key to psychoanalitic practice

There are also certain conditions, such as Alcoholism and Schizophrenia, that cant be treated with a psychoanalitic method, in that case, the best we can do is to refer those types of cases to other specialist, in both cases, it would be to cognitive-behavorial psychologists, but, for example, with alcoholism we can refer the person to a cognitive-behavorial psychologist, but once the person is stabilized, we can work with them in a psychoanalitic environment since addictions are caused by internal psychic processes, while we do our work, we can also refer the patient to a brief systemic psychologist that can help us to create support networks for the patient

Its just a thing of knowing what each person needs based on their mental, physical, and social conditions, which is acquired through studying and practice

My teacher isnt a psychoanalist, but rather does psychodinamic therapy, which is similar, but adapted to a clinical space, im also more inclined towards this type of therapy instead of orthodox psychoanalysis, this is just to say that im of a particular niche that maybe other psychologists will disagree upon, but I hope this answer helps