Everyone is a server is pretty much the opposite of "serverless". The article is just illustrating the migration path from traditional web assets from centralized servers and how you can use them in the distributed web.

A real life use case would be something like:

You want to build a financial tracking app in the distributed web. But your bank is the central source for your data. With the method shown in the article you could request the api endpoint from bank.com inside your p2p/Beaker app then load that data into local storage/a dat archive/memory and use it to track your data. Think mint but without giving your bank credentials to a 3rd party.