In this talk, we present a novel convergence analysis for the numerical approximation of axisymmetric mean curvature flows, covering both isotropic and anisotropic cases for genus-1 surfaces. Our work directly addresses a key limitation in the analysis by Barrett et al. (2021), namely a restrictive time-step condition. We introduce a new time-space error splitting technique that successfully removes this constraint for the isotropic flow. Furthermore, we demonstrate that this technique provides a crucial tool for the more complex anisotropic case. It helps overcome a fundamental obstacle: the inability to guarantee the boundedness of the solution's gradient, which is essential for the convergence analysis. Our unified framework shows that the analytical strategy developed for the isotropic case is powerfully applicable to the anisotropic setting, ensuring the required properties for convergence. The talk will conclude with numerical experiments that validate our theoretical findings and illustrate the dynamics of the flow.