We revisit the Partially-Saturated-Cells method (PSC) in the context of incompressible single-phase flows past stationary and moving bodies. This methodology adds a solid collision operator to the Lattice Boltzmann Equation, resulting in a unified evolution equation everywhere in the domain that suitably accounts for the boundary conditions. The unified evolution equation, which also allows for a smooth transition from solid cells to fluid cells, is dictated by the solid volume fraction, obtained using a simple geometric approach. We prove that the PSC approach exhibits nominal second-order accuracy in presence of straight and curved boundaries for pressure, although spatial order of accuracy in velocity degrades to first-order. Our analysis also reveals that PSC is approximately discretely conservative in the incompressible limit with the mass conservation errors being dependent on the lattice size and relaxation time. For simulations involving moving body problems, the method is found to introduce acceptable level of spurious force oscillations, which rapidly decrease with increasing grid resolution. To the best of the authors' knowledge, the present investigations pertaining to discrete conservation and spurious force oscillations in the PSC approach have never been discussed in literature. Extensive numerical experiments are performed involving both stationary and moving bodies, with imposed and induced motion, to assess the applicability of the PSC approach. Our findings illustrate that the inclusion of added mass effects are important for accurate force and torque computations in case of accelerating solids. Numerical investigations, that include unsteady flows featuring three degrees-of-freedom fluid-particle interactions clearly highlight the ability of the PSC approach in resolving complex hydrodynamical phenomena. This in-depth evaluation of the PSC approach conclusively demonstrates its efficacy as a promising computational framework for efficient and robust computations of incompressible fluid flows over a range of flow scenarios. © 2022 Elsevier Ltd