Thin sheets can be assembled into tubular origami structures that combine deployability with pronounced anisotropic stiffness, enabling applications ranging from robotics to deployable systems. Most existing tubular origami designs rely on degree-four origami vertices, whose single degree-of-freedom kinematics are well understood and relatively straightforward to design. As a result, tubular architectures based on higher-degree vertices, which possess greater local kinematic freedom, have remained largely unexplored due to the perceived difficulty of achieving stiffness and controllability. Here, we introduce a systematic and automatable design framework that enables the generation, evaluation, and optimization of tubular origami structures based on generalized degree-n vertices. Using this framework, we perform a comprehensive exploration of the design space and demonstrate that increased local kinematic freedom does not compromise stiffness at the structural level. Instead, through geometric coupling in tubular assemblies, higher-degree vertices can produce globally stiffer and more anisotropic architectures. Numerical simulations, validated by experiments, are used to characterize the large-deformation stiffness response and to formulate objective functions that quantify directional stiffness, including axial, radial, and rotational components. Our results reveal two key outcomes. First, systematic optimization of degree-four vertex geometries yields up to a 20% improvement in anisotropic stiffness compared to state-of-the-art tubular origami designs. Second, replacing the unit cell with higher-degree vertices further amplifies anisotropy, achieving improvements of up to 40% relative to existing designs. These findings establish high-degree origami vertices as a powerful and previously underutilized design element and demonstrate that automated, framework-driven exploration provides a new route for tuning anisotropic stiffness in tubular origami beyond conventional design paradigms.