I didn't try Zed but VSCode does the same. It is automatic, so you may not realize that it is installing a server, but it does. That it only needs SSH means that it doesn't need an additional communication channel, but you still need to be able to install something. In practice, it means you need a typical non-root shell account.

But Sublime Text can't do that, it is not a client-server application, everything has to run on the same machine, while Zed and VSCode can split the UI part and file access part on two different machines.