On-policy distillation (OPD) trains a student on its own trajectories with token-level teacher feedback and often outperforms off-policy distillation and reinforcement learning with verifiable rewards. However, we find that its standard advantage-weighted policy gradient suffers from three structural weaknesses, including high-variance updates, vanishing gradients in zero-advantage regions, and exploration bottlenecks for models with limited capability. We therefore propose asymmetric on-policy distillation (AOPD), which replaces ineffective policy gradient with localized divergence minimization in non-positive advantage regions while preserving positive reinforcement learning. Experiments on mathematical reasoning benchmarks show that AOPD consistently outperforms standard OPD, with average gains of 4.09 / 8.34 under strong / weak initialization, respectively. AOPD also maintains higher policy entropy during training and better capability retention during sequential tool-use adaptation. Our source code is available at [repository](https://github.com/stargazer319/AOPD).