By telling users you have better privacy simply because your browser adds a random header tag onto requests you're misleading them. Sites have no obligation to obey it and it will give users a false sense of security.
We already have proven, well defined method of DNT already: private browsing. This works by simply removing resources after the session ends. There's no reliance on servers, so it will always work.
Until appropriate legislation comes into play. It worked reasonably well with the Do Not Call lists.
Catching up is a matter of when, not if.
A number of large ad networks have already announced support for DNT. Sure, there will always be sites that ignore the header, but there's also a realistic chance that it will give users a meaningful choice about how their behavioral data is used by many of the biggest sites they use most often.
It's a lot harder to detect, and probably harder yet to prove, when they're engaging in illicit tracking.
It's the U.S. government trying to take anonymity and privacy away from the Internet; do we want them making weak, unenforceable, irremovable laws on how to implement an HTTP header?
You are still being tracked with private browsing. You IP, user agent, OS, screen resolution, etc. are enough to connect the dots in most situations.
This is a decent idea that will need to be implemented to some degree if sites are eventually going to support under-13s, since it is illegal to track their behavior under current law. But you are right, I don't see a lot of places voluntarily signing up to make less money.
And by session, this wouldn't just be the cookie store, it would also be the cache store and everything else which can be manipulated for tracking purposes.
EDIT: I'd also want it clearing between browser restarts too.
If this became (over night) the default standard way for browsers to behave, the vast vast majority of websites (at least 99.9% I'd guess) would continue to work without issue and it would pretty much annihilate nearly all privacy/tracking issues on the web.
Stuff like Google Analytics would continue to be able to work, but wouldn't be able to gather as many different types of information. Advertisers would be able to continue to advertise, but they would lose the ability to track you across sites, which potentially might hit their revenue a bit.