On-policy distillation (OPD) can improve answer accuracy while weakening responsiveness to supporting evidence. We investigate this separation in scientific-document question answering through matched document interventions that change the supported answer while preserving the question and unrelated context. We identify evidence bypass: a student retains an originally correct answer after the document changes to support a different one. On QASPER with a Qwen3-4B student, standard OPD raises original-document accuracy by 3.4 percentage points, lowers counterfactual accuracy by 5.3 points, and increases bypass by 15.7 points; the same directional pattern appears on PeerQA. Controlled perturbations localize the shift to answer-critical evidence, and bypass grows with the student's pre-distillation closed-book preference for the original answer. A factorized study of teacher targets, privileged information, divergence, and supervision density finds that explicitly supplying the teacher with annotated answer-critical spans yields the largest joint improvement in evidence response and claim support. These findings motivate Evidence-Paired OPD (EP-OPD), which combines on-policy supervision on matched documents supporting different answers to the same question with an answer-level margin that favors each document's supported answer over its paired alternative. Across both datasets and two student sizes, EP-OPD recovers 54-66% of standard OPD's counterfactual-accuracy loss while preserving 80-85% of its original-document accuracy gain. Our results establish evidence responsiveness as a distinct training and evaluation target alongside answer accuracy. Our code is publicly available at: https://github.com/RuoranXu/EP-OPD.

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