The present paper aims to clarify the mechanisms of the good numerical stability of the lattice Boltzmann model for incompressible flows and then construct a macroscopic solver that combines the good numerical stability of the lattice Boltzmann model and the flexibility of macroscopic algorithms in using nonuniform meshes. By introducing the second-order non-equilibrium momentto make entropy increase approximately in the collision process, we propose the non-equilibrium-moments-based reconstructed macroscopic equations of the lattice Boltzmann model for incompressible flows. Numerical investigations demonstrate that the discretized macroscopic equations can achieve even better numerical stability than the lattice Boltzmann model. It implies that the good numerical stability of the lattice Boltzmann model can be roughly explained by the entropy increase of the collision step and the diffusion process of the streaming step. However, the discretized non-equilibrium-moments-based reconstructed macroscopic equations are proven to have significant numerical errors at small kinematic viscosities. By combining them with the equilibrium-moments-based reconstructed macroscopic equations, a hybrid model that can achieve good numerical stability and accuracy at both small and large kinematic viscosities is proposed. Compared with the lattice Boltzmann model for incompressible flows, the hybrid model has better numerical stability, similar accuracy, and a significant advantage in using nonuniform meshes.