The thing is, though, the subset of vi you have to learn to is relatively small. You need to know how to enter and leave insert mode, how to delete, and how to quit with and without saving. (For a non-vi user, the difference between knowing and not knowing those simple things is significant)
This is generally enough to edit the few configuration files you may need to touch before you install emacs (network, sources.list, sudoers, etc).
So even though the use case is small and shrinking, the amount of "vi skill" you need is pretty low as well. There's really no good excuse to avoid it.
As a self-learner out in the wilderness, I don't know how I could have learned Linux without learning that basic Vi alongside. And that really is all you need to know.
However, it's more likely that you will have some version of Vi. The one I'm thinking of is the Busybox version.
That is enough justification, especially when you're at a single user mode console on a ILO card when the machine won't boot and flames are coming out of everything. Vi is there for you and emacs SSH won't be saving you when the network is not up so you cant apt-get or yum emacs. It's also not the sort of time you want to start having to learn vi.
I know this because I've been there.
It's as important as it always was, much as pen and paper are.
Emacs used to be considered 'expensive' in terms of disk/memory. These days, it really isn't.
Non-Linux unixes are less and less of a factor these days, and even most of those have some sort of packaging system where installing emacs is a quick operation, rather than a laborious download/compile/install.
Emacs itself has remote editing capabilities with Tramp, via ssh that obviates the need to fire up an editor on the target machine in some cases.
If your job involves sitting down at HPUX/Irix/AIX/whatever machines that haven't been updated since 1998, and don't allow remote access, yes, vi is probably a valuable skill. Otherwise, I think this argument is less important than it once was.