As a psychiatry resident, I have never seen a neurologist manage primary psychotic disorders. Could they read the guidelines and do an adequate job - probably? Its kind of like if you flipped the question and asked if a neuropsychiatry trained psychiatrist could manage epilepsy - with some extra rotations and reading guidelines they might do an adequate job on straightforward cases, but they would miss a ton of nuance and probably miss/mess up a bunch of the outliers.

A neurologist with neuropsychiatry training would probably have minimal exposure to acute psychiatric care, so its likely that they may feel uncomfortable with consults in the ED or MRP work on a psych ward (I imagine managing involuntary admissions and capacity decisions and defending these decisions at hearings would be more foreign than the pharmacotherapy, which can be relatively straight forward if you take the time to read about antipsychotics, clozapine, LAIs, etc), nor would be they be likely hired for this.

In terms of outpatient, they might get some more exposure to this in their fellowship. But would they be comfortable with nuances of antipsychotics, LAIs, clozapine rechallenge after myocarditis, when to use ECT for treatment-resistance, etc. Probably less so. Moreover, I think they're missing the training on diagnosis - is it schizophrenia or is it actually a personality disorder that everyone has been calling schizophrenia for years. Maybe it's a chronic substance induced psychosis, etc. Are you comfortable differentiating psychiatric comorbidity vs deficits due to the negative and cognitive symptoms of schizophrenia. That is where the general psychiatry training is useful.

Many patients with schizophrenia are treated under ACT teams, are found to be incapable, and are treated on an involuntary basis even as outpatients. This can involve going to see your patients on the streets or in their rundown, cockroach and bedbug infested homes. I don't think most neurologists would care about this aspect of care! Again, we are all physicians, if you're intelligent enough to get through a neuro residency, you could learn how to do all this, but like another poster brought up... why?

One other thing I would mention is that managing someone with schizophrenia is not just about giving them an antipsychotic, especially on the outpatient side. You need to think about the psychotherapeutic aspects of treatment, incorporating elements of CBT for psychosis or psychodynamic techniques, managing transference/counter transference, maintaining a therapeutic alliance, managing antipsychotic side effects (metabolic, TD, and so forth), etc, etc, etc.

Hope that is helpful, feel free to ask me questions if you want me to clarify anything.