Crystalline Bi2Se3 is one of the most explored three-dimensional (3D) topological insulators with a 0.3 eV energy gap making it promising for applications. Its amorphous counterpart could bring to light new possibilities for large scale synthesis and applications. Using ab initio molecular dynamics simulations, we have studied realistic amorphous Bi2Se3 phases generated by different processes of melting, quenching, and annealing. Extensive structural and electronic characterizations show that the melting process induces an energy gap decrease ruled by growth of the defective local environments. This behavior dictates a weak stability of the topological phase to disorder, characterized by the spin Bott index. Interestingly, we identify the occurrence of topologically trivial surface states in amorphous Bi2Se3 that show a strong resemblance to standard helical topological states. Our results and methods advance the search of topological phases in 3D amorphous solids.