Consider engineered mice:
“Nude” mice lack a thymus and hence make no T cells. They are “immunodeficient,” but not severely so. Yet by definition they are more immunodeficient than someone whose immunodeficiency is solely due to HIV (CD4- after prolonged infection, but still make other T cells, dendritic cells, B cells, and have innate immunity and cytokine activity).
SCID mice lack T cells and B cells. They are severely immunodeficient, but do have innate immunity.
NSG mice are SCID mice with most of their innate immunity knocked out. They lack complement, have defective macrophage activity, reduced natural killer cells, but do have granulocytes, and most of their cytokine activities (minus 6 interleukins), so not completely immunodeficient either.
Consequently, if the mouse model system is representative of the human case, severely immunodeficient people with HIV have at least one other significant immune impairment. Because less than 1% of health care workers, few of whom are immunodeficient, who accidentally get a needle stick with HIV-infected blood get AIDS, it is likely that multiple deficiencies in defense systems, including of course immunodeficiency, precede HIV infection, and allows it to take hold, and over time, HIV makes the person more immunodeficient.
What would the infection rate be without the quick medical attention so many medical professionals receive? Don’t know – but it would not be nearly 100%, the number expected if HIV were THE CAUSE of AIDS.