I know this post is a few months old now however I needed to chime in.
The position of network administrator is hardly rare or endangered. Obviously the majority of your experience is with smaller companies, where as you mentioned the limited IT staff usually does everything.
However all large corporations have many network administrations. It is not uncommon for there to be multiple daily changes to the network, there is no "set it and forget it" when you are talking about a large global corporation.
When a company has over 50,000 WAN routers (not even counting local routers and switches, wireless access points, etc.) they need large teams of dedicated Network Administrators.