Unraveling the cosmological Li problem - the discrepancy between Big Bang nucleosynthesis predictions and observed values in the Spite plateau - requires a comprehensive exploration of stellar evolution. In this study, we utilized the code STAREVOL to compute the stellar evolution models with atomic diffusion, rotation-induced processes, parametric turbulence, and additional viscosity. We calibrated the models to fit the abundance of Li in Population II stars selected from the GALAH DR3 spectroscopic survey and literature compilation based on their chemical composition. The calibration reveals the significance of parametric turbulence in counteracting atomic diffusion effects. These models predict the constancy of the Spite plateau as a function of $T_\mathrm{eff}$ and [Fe/H] which agrees with the observational trend found after a detailed selection of dwarf non-peculiar stars. Other dwarfs that lie below the Spite plateau are either CEMP or have other types of chemical peculiarities, reinforcing the notion of their environmental origin. The Li abundance near the Spite plateau of the most Fe-deficient star, J0023+0307, which is not CEMP, provides additional evidence for the stellar depletion solution of the Li cosmological problem. Also, our models predict a transition from Li constancy at low metallicities to dispersion at high metallicities which is seen in observations. In addition, we extend our analysis to include a comparison with observational data from the globular cluster NGC 6752, showcasing excellent agreement between model predictions and Li and Mg trends in post-turnoff stars. This opens avenues for refining the estimates of initial Li abundance in metal-rich globular clusters which would help to constrain Li evolution in the Milky Way.