I wouldn't want to noodle on this too much. I take your point. Anycast abstracts away some of the notion that you're talking to a specific computer. But the unicast service model is inherently about computers talking to each other. Many, maybe most, of the most important Internet applications aren't about 1-1 conversations, or if they are, they're 1-1 conversations in special cases of systems that also work 1-many.
One opinion is that a very important Internet application will inevitably be 1-1.
Who did the FCC just hire as their new CTO? What is happening to POTS?
1-to-many systems, hacked to give an illusion of 1-1 conversations, e.g. smtp middlemen, social networking's http servers or twitter-like broadcast sms, are what we settle for today, but, imo, this is a limitation not a desired goal.