While it is true that one does not need a C-L fellowship to practice in C-L settings, I disagree that doing consults is just bread and butter psychiatry like seeing medicine or neurology consults. The reason C-L is its own subspecialty is because the work is in many cases quite different from what the average psychiatrist is doing. This contrasts with medicine consults or other specialty consults where the work is exactly the same as what they might be doing in other settings with some notable exceptions (e.g. medicine consults for perioperative management of patients). There is a whole body of knowledge and skill set that goes along with C-L psychiatry, especially in larger hospitals (where academic or non-academic) that most general psychiatrists do not get.
One of the skills in C-L that differs from consults in other specialties is a good C-L service is providing consultation related to teams, and takes into account systemic dynamics, systems based practice, medicolegal issues, and ethical issues. Also, a really good C-L service will integrate things like brief psychotherapy, behavioral interventions, hypnosis, mindfulness etc into care of patients.
For example being really confident with psychopharmacology in medically complex patients, management of delirium, somatoform disorders, factitious disorders, transplant patients, HIV psychiatry, psycho-oncology, reproductive psychiatry, mind-body medicine, neuropsychiatry, chronic pain, addiction in medically complex patients etc. These are things most general psychiatrists are not good at.
I did not do a C-L fellowship but I do general C-L and neuropsychiatry consults both outpatient and inpatient. You can learn stuff on the job (and for some people it will be much better to do this), but for some people doing the fellowship provides the structure, mentorship and experiences to get them where they want to be. No, it's probably not going to make you more money. It's probably not going to make you more employable or be necessary at a community hospital. However, working at specialty hospitals or academic settings, C-L psychiatry is its own subfield which requires a higher knowledge base and skillset to do well compared to other areas of psychiatry. Also for those interested in administative medicine, leadership positions and medical education, you will note that C-L psychiatrists are overrepresented in such positions.
I second the comment about going somewhere that has a really good fellowship program or else there's probably not much point. Programs that have historically been strong in this include MGH, Brigham and Women's, Columbia, MSKCC, Stanford, UW, Inova Fairfax
Also even in academic settings, C-L fellowship is not required. I know of no institutions that require one to have such a fellowship to work on the C-L service. However other fellowships are also desirable e.g. addictions, forensic, geriatrics, palliative medicine, even child psych. The best C-L divisions have people from different specialty areas.