The development of landslide early-warning systems is of paramount importance for cities located in mountainous areas. Landslide dynamic encompasses various velocity scales and mechanical behaviors, presenting challenges in numerical modeling. In this work, we focus on landslide runout phase. While the comparison of the horizontal propagation length scale of the landslide with the vertical one suggests the use of depth-integrated models, the different behavior of various types of landslide, such as debris flows and mudflows, entails the need to define more detailed mathematical models. Two numerical frameworks are proposed: one for homogeneous moving slides, such as mudflows, and the other for landslides with significant solid-liquid interactions. These schemes adopt adaptive mesh refinement and domain partitioning on hierarchical quadtree meshes, enhancing scalability and efficiency. These numerical schemes can be seen as modifications of the standard two-step Taylor-Galerkin (TG2) scheme on quadtree meshes. We propose modifications aimed at ensuring well-balancing property and overcoming the numerical stiffness coming from diffusion-reaction terms, while guaranteeing scaling performances proper of the TG2 scheme. To ensure the well-balancing property we resort on a novel implementation of the path-conservative strategy in TG2 scheme and employ either second-order operator splitting methods or implicit-explicit additive Runge-Kutta scheme to face numerical stiffness coming from the finite-element discretization of diffusion-reaction terms. Numerical experiments demonstrate the effectiveness of the schemes in terms of time-step selection, parallel performance, accuracy, and ability to deal with real topographies.