It's a client to a client-server application.
Still, there are tradeoffs to what you're stating. You're complaining because of your network topology. But consider what happens when your servers are firewalled against one another. Or if they're access controlled against one another.
If you want to do what you're stating, that's also not particularly hard. It's a run-on-server agent that copies all files. And you ... run it on the server.
So: you have both options.
Aside, the Administrator is capable of moving databases onserver. That's a function that's inaccessible to the other clients as far as I know.