In this article we prove highly improved and flexible Strichartz-type estimates allowing us to generalize the asymptotics we obtained for a stratified and rotating incompressible Navier-Stokes system: for large (and less regular) initial data, we obtain global well-posedness, asymptotics (as the Rossby number ε goes to zero) and convergence rates as a power of the small parameter ε. Our approach is lead by the special structure of the limit system: the 3D quasi-geostrophic system.