I don't get how this would work. For someone logging in you still have to talk to a central database holding the authentication information. Really anything with a database has a similar problem if you want to enforce consistency.
To use the subdomain example, say you want to prevent two people from reserving the same name. That means the reservation from someone in Sydney has to propogate to the worker in NYC before you can confirm that it has succeeded. Otherwise, simultaneous reservations will result in two successful reservation. So you've lost the benefit of edge workers.