Most STDs we don't have vaccines for, and the ones that have survived to the modern day are the ones that can be good at hiding. Syphilis, for example, has the nasty habit of seemingly going away by itself, only to pop up again years later. Chlamydia and Gonorrhea can be asymptomatic.

The only reason we haven't eradicated TB is because we've failed massively in getting the vaccine to people in the poorer parts of the world. We could eliminate it just like we did smallpox, the only barrier is actually doing the damn thing. It is made more difficult by the fact that it has a rather long latent infection time, but again, could be solved by vaccinating everyone.