This paper analyzes the global dynamics of an Human Immunodeficiency Virus (HIV) delay model which incorporates cytokine enhancement and three saturated incidence rates. Based on the distinct thresholds of two reproduction numbers, we establish the existence of both immunity-inactivated and immunity-activated equilibria. Furthermore, the global attractivity of all three equilibria is rigorously established through constructing Lyapunov functionals. Our simulations demonstrate the following: (i) enhanced virus-cell saturation demonstrates a superior efficacy over the saturation effect of inflammatory cytokines in driving systemic parameters toward Acquired Immunodeficiency Syndrome (AIDS) amelioration; (ii) immune saturation can critically impair anti-HIV defense mechanisms; (iii) increasing the virus-cell saturation levels significantly neutralizes the adverse effects of immune saturation on disease progression; and (iv) time delays exhibit therapeutic benefits within an optimal range, with diminishing returns beyond this threshold. These results suggest both saturation parameters and time delays represent potential therapeutic targets for HIV treatment.