Not really. The only substantial change is that getting thrown out of such a large and all-encompassing diner is a lot less convenient. The fundamental reality is still there: no website (maybe unless it's owned by your government, and even then) no matter how large is obligated to carry your message, and it is increasingly affordable and trivial to start your own website if other websites exercise their inherent rights of refusal to carry your message.
> Couldn't you use this formulation to say that anti-discrimination laws make it so that, simply by dint of providing a service, you take on an obligation to provide that service to everyone?
The obligation is to not make membership in a protected class or lack thereof a condition of providing a service. Bob can't throw you out of his diner on the basis of you being some race he doesn't like; he can nonetheless throw you out if you're shouting advertisements at everyone else in the diner. Same deal for a website. There is no implication there of any obligation to serve everyone: only an explication of constraints on the reasons someone can refuse to serve someone.
Marketers are, in short, not a protected class.