Does anyone think this is a bad idea?
Yes, someone does. Organizations like the Family Research Council, the Abstinence & Marriage Education Partnership, and other sexual conservatives think that vaccinating minors against a sexually transmitted disease will encourage promiscuous sex. From their point of view, HPV infection only affects sexually active women with multiple partners and gay men. HPV is also their great example of why condoms "don't work", because it can be spread by skin contact other than the penis itself. So, no HPV problem means that condoms are 100% effective; can't have that.
Some pretty rich quotes from the FRC are in this article from New Scientist.
So, here we have a disease that kills thousands upon thousands of people a year, and causes incredible amounts of fear and pain even when it doesn't kill. It's spread by a virus. We have a vaccine that wipes it out. And these people don't like it because it might encourage extramarital sex among teenagers. Because to their mind their sky god has told them that sex outside of marriage is worse than death.
This why I am no longer a Christian. And why I am not the agnostic I was before Christianity, but a thoroughgoing atheist. This kind of behavior outweighs any good that may result from spirituality. Look, you can do what you want for your religion: wear 17th century clothing, refuse military service, eat a restricted diet, carry a little knife everywhere, wear magic underwear. But if you tell me that a quarter of a million people a year need to die for your abstraction you are my mortal enemy. I'm really uninterested in your arguments.