Combining lattice model simulations with a renormalization group (RG) analysis of effective theories, we investigate the nature of the Anderson transition in chiral symmetry classes with one-dimensional (1D) weak topology. In the simulation study, we extend previous transfer matrix analyses to the chiral symplectic class, and study numerical Lyapunov exponents via a finite-size scaling (FSS) analysis that assumes spatially isotropic scaling. The analysis shows that, as in the other two chiral symmetry classes, the weak topology induces an intermediate quasi-localized (QL) phase between metal and Anderson insulator phases. In this QL phase, the localization length of wave functions diverges exclusively along the direction of the 1D weak topology. A comparison of the universal critical exponents across the three chiral symmetry classes indicates that the quantum criticality of the metal-to-QL transition varies, depending on the nature of the time-reversal symmetry. Motivated by this numerical observation, we revisit and extend our previous two-dimensional (2D) RG analysis to all three chiral classes, now newly incorporating a one-loop renormalization of the weak topological term in the analysis. The revised analysis reveals that a quasi-localized strong-coupling fixed point previously reported in the chiral unitary class is unstable under this new inclusion; instead, the strong-coupling phase is entirely governed by a stable fixed point with conventional localized character. Nevertheless, in the chiral unitary and chiral symplectic classes, the RG analysis still yields the hallmark of the 1D weak topology through the spatially anisotropic scaling of the Anderson transition criticality. These theoretical findings suggest that the quasi-localized phase observed numerically in 2D models may be an artifact of the spatially isotropic scaling assumption in the FSS analysis. A conclusive numerical identification of this phase therefore requires a finite-size scaling approach that accommodates generic (anisotropic) spatial scaling.