As demand for mental health care outpaces clinician-delivered assessment, scalable screening tools are increasingly needed. Large language models (LLMs) may identify psychiatric risk from patient narratives, but their reliability across diagnoses, demographic subgroups, and evidence-use patterns remains uncertain. We introduce a SCID-anchored benchmark of 555 semi-structured experiential interviews paired with diagnostic reference labels for anxiety disorder, major depressive disorder, post-traumatic stress disorder, and any current mental health disorder. Using zero-shot task-specific prompting, we evaluated five state-of-the-art LLMs and examined whether false-negative errors reflected missed psychiatric evidence or differential weighting of symptom, functional-impairment, and protective-context cues. Performance varied across tasks and models, with accuracy ranging from 0.49 to 0.86 and Matthews correlation coefficients from 0.16 to 0.38. GPT-4.1 Mini and GPT-5 Mini showed the most consistent disorder-specific accuracy. Subgroup analyses found higher depression-classification accuracy among male than female participants, no consistent age-related pattern, and modest non-uniform variation across race strata. Evidence-integration analyses showed that false-negative anxiety and PTSD classifications often contained explicit symptom evidence but were accompanied by preserved functioning, coping ability, or social support. Functional-impairment evidence shifted model outputs toward positive classifications, whereas protective-context evidence shifted outputs away. These findings suggest that LLMs may support scalable psychiatric screening, but their tendency to discount symptom evidence in the presence of preserved functioning or protective context requires careful validation before clinical deployment.