Significant progress has been made in video restoration under rainy conditions over the past decade, largely propelled by advancements in deep learning. Nevertheless, existing methods that depend on paired data struggle to generalize effectively to real-world scenarios, primarily due to the disparity between synthetic and authentic rain effects. To address these limitations, we propose a dual-branch spatio-temporal state-space model to enhance rain streak removal in video sequences. Specifically, we design spatial and temporal state-space model layers to extract spatial features and incorporate temporal dependencies across frames, respectively. To improve multi-frame feature fusion, we derive a dynamic stacking filter, which adaptively approximates statistical filters for superior pixel-wise feature refinement. Moreover, we integrate a median stacking loss to enable semi-supervised learning by generating pseudo-clean patches based on the sparsity prior of rain. To further explore the capacity of deraining models in supporting other vision-based tasks in rainy environments, we introduce a novel real-world benchmark focused on object detection and tracking in rainy conditions. Our method is extensively evaluated across multiple benchmarks containing numerous synthetic and real-world rainy videos, consistently demonstrating its superiority in quantitative metrics, visual quality, efficiency, and its utility for downstream tasks. Our code will be made publicly available.