Unfortunately such people are evidently few and far between.
Banning cookies and having the client offer a session identifier instead solves many problems.
For starters, it stores the data where it belongs: On the server, putting the cost of storage and protection where it belongs too.
This is a win for privacy, as you will know if you have ever taken the time to actually examine the cookies on your own machine.
Second, it allows the client+user to decide if it will issue anonymous (ie: ever-changing) session identifiers, as a public PC in a library should do, or issue a stable user-specific session-id, to get the convenience of being recognized by the server without constant re-authorization.
Today users don't have that choice, since they have no realistic way of knowing which cookies belongs to a particular website due to 3rd-party cookies and image-domain splitting etc.
Network-wise, we eliminate a lot of bytes to send and receive.
One of the major improvements SPDY has shown is getting the entire request into one packet (by deflating all the headers).
But the only reason HTTP requests don't fit in a single packet to begin with is cookies, get rid of cookies, and almost all requests fit inside the first MTU.
Finally, eliminating cookies improve caching opportunities, which will help both client and server side get a better web experience.
As for breaking the world: It won't happen.
It is trivial to write a module for apache which simulates cookies for old HTTP/1 web-apps: Simply store/look up the cookies in a local database table, indexed by the session-id the client provided.
I'm sure sysadmins will have concerns about the size of that table, but that is an improvement, today the cost is borne by the web-users.