Is it true a decent psychiatrist should exhaust the psychosomatic route first?

Your question is confusing. First of all, psychiatrists treat psychosomatic illnesses as part of their general practice. With regard to "exhausting physical and neurological reasons" you are absolutely correct. A "good" psychiatrist will do enough history to determine whether or not there might be some organic reason for certain symptoms. Generally this would involve a complete physical, including blood chemistry. A Neurological evaluation is also a good idea.