Retrieval-augmented generation systems often suffer from a gap between optimizing retrieval relevance and generative utility. With such a gap, retrieved documents may be topically relevant but still lack the content needed for effective reasoning during generation. While existing bridge modules attempt to rewrite the retrieved text for better generation, we show how they fail by not capturing "document utility". In this work, we propose R2U, with a key distinction of approximating true utility through joint observation of rewriting and answering in the reasoning process. To distill, R2U scale such supervision to enhance reliability in distillation. We further construct utility-improvement supervision by measuring the generator's gain of the answer under the rewritten context, yielding signals for fine-tuning and preference optimization. We evaluate our method across multiple open-domain question-answering benchmarks. The empirical results demonstrate consistent improvements over strong bridging baselines