Purpose: Surgical workflow recognition enables context-aware assistance and skill assessment in computer-assisted interventions. Despite recent advances, current methods suffer from two critical challenges: prediction jitter across consecutive frames and poor discrimination of ambiguous phases. This paper aims to develop a stable framework by selectively propagating reliable historical information and explicitly modeling uncertainty for hard sample enhancement. Methods: We propose a dual-pathway framework DSTED with Reliable Memory Propagation (RMP) and Uncertainty-Aware Prototype Retrieval (UPR). RMP maintains temporal coherence by filtering and fusing high-confidence historical features through multi-criteria reliability assessment. UPR constructs learnable class-specific prototypes from high-uncertainty samples and performs adaptive prototype matching to refine ambiguous frame representations. Finally, a confidence-driven gate dynamically balances both pathways based on prediction certainty. Results: Our method achieves state-of-the-art performance on AutoLaparo-hysterectomy with 84.36% accuracy and 65.51% F1-score, surpassing the second-best method by 3.51% and 4.88% respectively. Ablations reveal complementary gains from RMP (2.19%) and UPR (1.93%), with synergistic effects when combined. Extensive analysis confirms substantial reduction in temporal jitter and marked improvement on challenging phase transitions. Conclusion: Our dual-pathway design introduces a novel paradigm for stable workflow recognition, demonstrating that decoupling the modeling of temporal consistency and phase ambiguity yields superior performance and clinical applicability.