P3P also exposes a machine-readable semantics for privacy policies, which could be used by the browser to manage access to not just cookies but other problematic features as well. It's a lot more flexible than "just reject cookies" or the "DNT" header.
The issue is imho that the decision needs to happen server-side (so that a user with an incompatible client isn't slurped up by default), ergo the user-agent must declare the intentions of the user and the server needs to act on it.
P3P could be a hint for the the user-agent, but the user-agent would have to tell the server what level of tracking etc is acceptable to the user.
Otoh, we already have cookie-consent in browsers. Just don't accept cookies if you don't consent!