In [Dai et al., Multi. Model. Simul. 18(4) (2020)], a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory, based on which a linearized method was developed in [Hu et al., EAJAM. 13(2) (2023)] for further improving the numerical efficiency. In this paper, a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn-Sham model. Temporally, the convergence, the asymptotic stability, as well as the structure-preserving property of the linearized numerical scheme in the method is discussed following previous works, while spatially, the convergence of the h-adaptive mesh method is demonstrated following [Chen et al., Multi. Model. Simul. 12 (2014)], with a key study on the boundedness of the KohnSham potential for the all-electron Kohn-Sham model. Numerical examples confirm the theoretical results very well.