If you RTFA, for every 100 actual cases, it misses 7. That's 7 people who will think that they're HIV-free, and possibly spread it to others.
7% false negatives is a *terrible* number.
No it isn't. If everybody used this test, and the people who tested positive seek treatment, then you just reduced the transmission rate among the people who would not otherwise get tested by 93%.
I often run in to arguments like this when it comes to vaccinations as well. Many vaccines don't offer perfect protection. They just reduce the probability that a disease will spread in the population by a sufficient amount that you don't get major outbreaks, and thus the illness eventually dies out since it cannot spread effectively. More specifically, if new infections occur at a rate that is lower than the rate at which infected people are discovered and treated, the the total number of infections decrease. Since untreated HIV will eventually start showing symptoms, people will eventually get diagnosed, so the trick is to bring the overall infection rate down enough that you're more likely to be diagnosed and treated than you are to spread it to others.
Thus while highly preferable, it need not be perfect. If your measures to combat the illness cause a persistent decrease in the number of people infected, then the disease will eventually die out simply because it cannot spread.
As it happens that is also one of the reasons why you want everybody to have access to healthcare.