The idea of something like Gardidsil is plain old good common sense to reduce the incidence of diseases that could be much more than just irritating. Its the same exact issue with the school-age vaccinations. Requiring some relatively innocuous preventatives to keep from epidemics is a rightful function of government for the common good. No different than the purposes of the military.
The Opt Out clause gave parents, who were confident their 12 y.o was not sexually active, a reasonable excuse from the vaccination, because Gradisil was not needed in their case.
If you or your libertarian friends fail to grasp the concept, then you are simply being political.
I am pretty sure I have no libertarian friends, since I often post this line.
True Libertarianism always leads to Perversion.