Better than the FP thread, but not that great. How about if you specifically consider better approaches to solutions?
On one hand I mostly agree with you, but in the other hand I have a giant can of worms...
But first a detour on the scenic route. I think your premise is broken. The Internet has never been anonymous. Just because the resources to nail people were out of your reach does not mean it was impossible to find you. Even if you were making deliberate technological efforts to hide, you can eventually be found, one way or another. Perhaps the best counterexample of your premise is the Unabomber. Yes, not on the Web--but I think that was half because of the timing and half because he understood the lack of real technology-based anonymity. But he tried quite hard to stay hidden. And died in prison.
I want to stay with solution-oriented thoughts. I have thought tracks on two aspects.
First, about age checks (nodding at the story). I think the best approach would involve probing childhood memories. The trick would be distinguishing between firsthand experiences as remembered from a child's perspective and various forms of secondhand information. But I do think that there are real differences in how children think and that it would be quite difficult to fake the massive body of "atmospheric" data. The attempts to create fake childhood memories would either link to known data (such as history books) or break down in contradictions as previous answers were explored. In essence I am arguing that the time of your childhood creates a unique signature on your personality. (Note that location is also crucial. Other factors include immediate family.)
Second track is about proving you are human. I think the best approach has to be based on your links to other real humans. Kind of like tracing security certificates. So how to prove each link in the chain? I imagine a system based on joint timelines. It would start with your memories of an event that you shared with another person. Your version would be used to build a kind of quiz that the other person would try to answer based on their memories of the same event. Of course I now imagine it as an AI application, where the AI is also tracking the flows of information (to detect reverse answer fishing) and trying hard to provide the best possible distractors for each question about the event (and considering likely mistakes in your own memories). Actually, the lack of perfect agreement would be evidence in favor of a real human being because our memories are flawed and each of us remembers things from a slightly different perspective and with a different context. I don't think the answers could be absolute, but each step in the chain could have a high probability, and as long as you stay within the infamous six degrees of Keven Bacon you could have a strong basis for believing an identity is a real human being.
So how do we get to Reykjavík with the real and verified human beings?
"First you go west about 10 clicks and turn left in front of a red barn, then... No, that won't work. Go north until you get to... No, that's no good. Sorry, folks. We can't get there from here."
What I really mean is that the google won't do it unless there's a profit. Ditto all the other evil corporate cancers. And the governments? Just minor subsidiaries now, but not even capable of solving any real problems. Then there's the ever-present time problem. In particular, I ain't gonna live long enough to laugh at the flop.