Actually it would be even "easier" to do than that. Unfortunetely we would need everyone within an age group to all agree to do it. And you know someone would screw it up.
Here's the idea: If we could just get one generation (start early like 11 years old) to only have sex with their same age or lower, or with people who have only had sex with "virgin-connected"* people for the rest of their lives, then it would work.
*what I mean by this is, say you have sex, you're both virgins before, after you wouldn't be virgins, but you would be "virgin-connected" (I need a better word for this, but can't think of one, sorry.) In other words, there's absolutely no way you could have an STD (well pretty much) and nor could any virgin-connected people.
That's it, no testing would need to be done, just honesty between people. The whole generation wouldn't even need to participate, so long as you kept the pool clean and seperated from the other pools so to speak.
Some ways we might actually eventually start this might be membership cards that state you're clean and have severe fines if you bring any STDs into the pool and membership revoked. Along with documenting all your sexual experiences (so if STDs are introduced it can be traced and dealt with.) Maybe even have exclusive clubs for casual sex for people with this membership. Eventually the club will be prestigeous enough that everyone will have joined if they can, membership will be wide enough that people rarely if ever need to have sex with others outside the club and the people with the incurable STDs will eventually die off, eliminating the last of the STDs.
Make sense?