HPV can be spread by direct skin-to-skin contact, that's the problem. So you can practise safe sex and still contract it
, although the risk is still far, far higher through fluids.
There wouldn't be a direct correlation between everyone only having safe sex and a significant reduction in HPV, sadly, since most carriers are asymptomatic, but there would be some.
My main reason for pushing safe sex and sexual history is because in an age where sex is so casual (nothing wrong with this by the way, this isn't a moral judgement), the fall out is something that is affecting women far more than men. Men don't get cervical cancer and women are something like twice or three times as likely to catch a STD from a man than men are from women. This affects women's fertility far more than it does men. Men don't get PID scarring their fallopian tubes either.
If more women refuse to have sex with a man unless he practises safe sex and can provide a full, cleared sexual history (including HPV, which I'd like to see included in screening), more men would practise safe sex and the risk of STDs for women and the subsequent risk to their fertility, would reduce. It wil never be eradicated, but it will reduce.