We propose a novel object-mediated hand interaction system that enables real-time operation with everyday objects on wearable augmented reality (AR) devices. Despite recent advances, both commercial and academic hand interaction techniques remain constrained, typically requiring external hardware or depending exclusively on bare-hand gestures. Motivated by these constraints, we developed an offloading framework that integrates a high-fidelity transformer-based 3D hand reconstruction model with a dynamic gesture recognition network powered by gated recurrent units (GRU). This architecture ensures stable and accurate gesture recognition even during interaction with physical objects. To evaluate its quantitative performance, we collected a custom dataset based on a predefined gesture set, achieving 93.0% accuracy in 5-fold cross-validation. The complete system implemented on Microsoft HoloLens 2 operates at a real-time framerate, and we further analyze the latency of each step in our framework. Through this interaction paradigm, users can experience immersive and intuitive AR in everyday environments with minimal disruption to natural action behavior. Our projects are available at https://github.com/kaist-uvrlab/UnifiedHOInteraction.