The network card enables the database server to exchange data with the outside world. This includes far more than web servers, spreadsheets, loading jobs, application servers, and other data consumers. The database server is part of a large continuum of activity, much of which will center around maintenance, management, and even filesystem availability.
Little of this other traffic involves PostgreSQL directly. Much happens in the background regardless of the database and its current workload. Yet even one mishandled network packet across an otherwise normal driver can render the entire server invisible to the outside world or, in extreme cases, even lead to a system panic and subsequent shutdown. On a busy database server, network cards can handle several terabytes of traffic on a daily basis; the margin of error for such a critical piece of hardware...