If I understood it correctly, the third party would hold your ID etc, and would vouch for you on signup to services. Users would be anonymous/pseudonymous until a warrant was acquired by the police.
A third party would be better IMO than the government. Not sure I like it either way tho. Who is the third party? Breaches would be an issue as the other poster has said. And too much like big brother.
Now that people are used to that idea, I expect a government to say "Why don't you store the logs on our computers instead, to save money and be protected with military-grade security. We promise we will still go to the hassle of getting a warrant before reading from the database... Unless of course there's an emergency...".
Perhaps the reason they haven't said this yet is because government cyber-security agencies have already gained real-time unwarranted access to the ISPs' logs. I don't suppose a company could be sued for not correctly securing this data against the government, since the government could hide behind "national security" to block any evidence being admitted in a court case.
Spread the third parties among jurisdictions and make N large enough that no government has enough of the third parties under its jurisdiction to be able to coerce an identity reveal.