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New head canon thanks to OP:
the pokemon storage system doesn't have a traditional server/client paradigm. It's a state-run (or in the case that pkmn centers are not state funded but franchised, committee-ran) p2p federated network of open source clients. It would be possible then, for our player to run their own instance of the software that stores pokemon on their PC at home. They could choose whether or not they wanted to connect their instance with the main network of server instances. In the case that they choose not to, their pokemon would only be retrievable from their house. This is rather inconvenient for someone going on a continent-spanning adventure; the benefits of running the software at home actually completely disappear if you remember that our character is never home.
This could be supported by the fact that the same software being run on your computer (item retrival / mailbox) is run on the PCs in the pokemon center. But, it's also completely destroyed by the fact that until gen 7 you are explicitly logging into another person's computer.
pic unrelated
