Proactive medical dialogue requires an agent to decide what to ask from incomplete patient information. Existing information-seeking approaches commonly prioritize questions that most reduce diagnostic uncertainty, but this criterion overlooks an important property of medical diagnosis: different diagnostic errors can carry substantially different consequences. The most informative question may therefore differ from the one most valuable for the downstream decision. We propose Expected-Severity-Risk (ESR), a consequence-aware question-supervision objective that values each candidate by its expected reduction in severity-aware terminal risk. Because questions must be selected before their answers are observed, ESR marginalizes over possible answers using train-only population statistics. Its rankings are then distilled into a prefix-only language policy, requiring no teacher-side risk computation at deployment. Across three matched Qwen3-4B training seeds on DDxPlus, ESR reduces mean high-severity diagnostic miss from 0.0645 to 0.0455 (29.5% relative reduction) and improves mean diagnostic accuracy from 0.9123 to 0.9320 while requiring only 0.14 additional questions per dialogue. Fixed-budget analyses show that the distinction persists when question count is controlled, while a matched expected-0/1-risk student control further isolates the contribution of asymmetric severity weighting. These results support moving proactive medical dialogue beyond uncertainty reduction toward consequence-aware evidence acquisition.

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