At a fine-grained level, different aspects of that experience can be said to occur specifically on client or server. Each of those aspects can be constrained or manipulated by the respective property owner.
When it comes to preferences of the visitor for certain server actions (or inactions), one can only make a request. This isn't a grand moral point, or a technical one, but one of basic property rights and personal freedom. And such a request is what the DNT header signifies.
Likewise, when the server has preferences for certain user agent actions (such as running JavaScript or storing cookies) again it can only request that this occur since the user agent can typically disable JavaScript or cookies. This is what certain HTML metadata elements and the Set-Cookie header signify.
If visitors are unhappy with the behavior of a server, they can avoid it. In aggregate, such avoidance can become a significant market force. At the same time, a website that does no analytics for DNT visitors and has a high ratio of DNT visitors may also become less competitive and valuable over time. Both can feedback into respective preference consideration. This is ultimately the meager value of DNT. It (combined with adequate education) provides extra context data that can motivate through market forces an adjustment to web browsing norms.
Along the lines of "adequate education", the option in Firefox should read "Tell websites to restrict their tracking of me. __(Learn more.)__"
Practically, how would you know in advance if a server will respect your DNT preference without first visiting the site? Well, in real life, how do you know whether someone who invites you over for dinner won't serve you poison? One way is through trusted third-parties, but the market hasn't yet demanded such a service (and may never).