This work is motivated by the existence of mapping the extended theory of gravity with a standard energy-momentum tensor to general relativity with a modified energy-momentum tensor. We construct a modified anisotropic energy-momentum tensor from a standard isotropic energy-momentum tensor by adding a "geometrical correction"to precisely reproduce the Tolman-Oppenheimer-Volkoff equations predicted by the generalized Tolman-Oppenheimer-Volkoff (GTOV) model. This construction aims to calculate the moment of inertia (I) and tidal deformability (Λ) of neutron stars (NSs) within the GTOV model. Therefore, we can comprehensively investigate the role of each free parameter of the GTOV model in NS properties. Furthermore, through this construction we can also utilize physically acceptable stability conditions for anisotropic stars to constrain the physical range of each parameter value and investigate the existence of a correlation among the parameters of the GTOV model. Except for α, we find that the values of the GTOV free parameters can be limited to acceptable ranges. We also find that the parameters θ, χ, and β are correlated, and the parameter Γ→0. With these free parameter ranges in hand, we study the role of each parameter of the GTOV model in NS properties, including I and Λ. We also revisit the hyperon puzzle in NSs within the GTOV model. We find that the θ parameter plays a crucial role in controlling the NS maximum mass value. We also find that the threshold k2 peak for NS is k2≈0.167. Furthermore, if we use parameter sets with θ=-1, the mass-radius predictions are compatible with recent NICER data on PSR J0030+0451 and PSR J0740+6620.