OK, let's go through some of the obvious points:
(1) Lots of diseases are effectively spread by shaking hands. We haven't banned shaking hands. Many diseases can be spread without physical contact between people (airborne diseases, viruses that can survive on an exposed surface for years). Other than the cultural shame attached to sexually transmitted diseases, why are they different from ordinary diseases? Anthrax and ebola do not primarily spread through sexual relations, yet their victims still tend to end up dead.
(2) Wanting to get rid of the age of consent laws that presume consensual sex is rape is not the same thing as promoting promiscuity. Unlike the gay community in the 1970s, most of us don't want to be having anonymous sex with hundreds or thousands of little girls. We tend to be interested in relationships. Promiscuity is what is primarily connected to transmission of sexually transmitted diseases, not age or difference in age.
(3) Besides promiscuity, specific sexual behaviors are associated with some of the more dangerous sexually transmitted diseases. Much of the sexual behaviors that pedophiles would prefer to engage in would not involve, for instance, the transfer of bodily fluids, and as such some of the most dangerous diseases, such as AIDS, would be very difficult to spread in the absence of that behavior.
(4) Who is more likely to spread a sexually transmitted disease to a child: an adult who is informed on the subject, or another child who knows just as little as they do and is promiscuous with other children? an adult who by necessity does not consult a doctor, or an adult who can freely ask questions without fear of persecution?
(5) Why do you presume that the adult would be the vector bringing disease to a child, rather than the other way around? Jobs in which adults care for children are notorious for putting adults at risk for many diseases that they catch from the children in their care. Why focus on the danger of adults infecting children, and not the danger of children infecting adults?
Of course disease is a concern that we should take seriously, but this is not a reason for restricting relationships on account of age.