Global reliability sensitivity (GRS) has received extensive attention and research because of its ability to reflect the contribution of each random variable’s variability to the system response variability and evaluate the effect of interactions between random variables on the model output variability. However, the high computational cost of GRS limits its use in engineering applications. To improve the practicality of GRS from an engineering perspective, this paper proposes an extensible GRS method based on radial-based importance sampling and adaptive surrogate models. First, a new radial-based importance sampling method is proposed which achieves adaptive updating of the hypersphere radius by sorting the adaptive surrogate model’s candidate samples twice. Secondly, a direct radial-based sampling algorithm is presented to enable rapid sampling outside the hypersphere and ensure the efficiency of reliability estimation. Secondly, a technique for reusing training sets is proposed to enable the failure probability to be calculated quickly in the Sobol index evaluation process. Finally, the effectiveness of the proposed algorithm is validated using two typical reliability sensitivity analysis cases.