ICD-10 codes are alphanumeric codes used in the US to classify diagnoses and injuries for medical billing and epidemiological reporting. Standard ICD-10 benchmarks report aggregate metrics that obscure performance on complex coding scenarios. We evaluate neural, workflow, and agentic systems on a rarity-stratified set of MIMIC-IV discharge summaries and identify two orthogonal failure modes. Neural classifiers exhibit a 0.43 micro-F1 gap between rare and common codes. Workflow systems handle rare codes well but score near zero on injury and external cause codes that require multi-step guideline following. A tool-augmented agentic configuration with structured access to official ICD-10 reference materials recovers up to 0.32 micro-F1 on this subset. No single system dominates across all conditions.