Achieving both efficient pushability (propulsion transmission) and proximally concentrated bending for steerability is challenging for magnetically actuated soft catheters: higher axial/bending stiffness improves force transmission but reduces steerability, whereas lower stiffness enables large, proximally concentrated bending yet increases kinking/buckling risk under compressive push loads. To address this trade-off, we propose a stiffness-optimized multi-segment magnetically actuated catheter (SO-MAC) that integrates a decoupled steering-advancement mechanism with a gradient-stiffness architecture. The SO-MAC concentrates bending about a stable proximal pivot during advancement while the distal section passively self-straightens to transmit propulsion, aided by the optimized stiffness distribution and elastic recovery of the spring backbone against friction-induced kinking/buckling. Over $0{-}180^{\circ}$ combined steering and advancement, the pivot remained stable and the distal tip advanced near-straight toward the target direction. A 1.5 mm-diameter SO-MAC achieved up to $180^{\circ}$ steering with a 3 mm bending radius at its 10 mm tip, with an average shape error of $1.39 \pm 0.56$ mm and a steering-pivot error of $0.35 \pm 0.10$ mm. Visual feedback control in a bronchial phantom further confirmed robust navigation through highly curved, bifurcating paths.