[1] http://www.gnu.org/software/emacs/manual/html_node/emacs/Rem...
There was a suggestion on a Reddit thread to use Dired instead: C-x d /ssh:me@remote: which I also tried. This appeared to get further but subsequently failed with the error "ssh_askpass: exec(/usr/lib/ssh/ssh-askpass): No such file or directory".
So I figured I'd try launching Emacs from within a Git Bash prompt to see whether that would provide the extra commands it was looking for, and it failed to locate my dotfiles (must have a different $HOME in that context...). However, when I attempted to open a remote file it did actually prompt me for a password! ...but then the LDAP authentication failed after timing out.
I'm out of gumption for now, but hopefully that helps to illustrate why I still want to be able to SSH into a machine and launch Emacs from the terminal.
I think it's great that the capability exists and that others find value in it; it's just not for me (or maybe vice versa). One of the great things about Emacs is that it can still work for both sets of people: folks who like TRAMP and folks who like to SSH in and run it locally.
I routinely SSH to my home desktop, where an instance of Emacs is always running, to work from e.g. a bus, or a queue to the doctors', etc. Since I'm normally working on my desktop, which runs GUI Emacs, when I SSH in I have all my files and processes opened as I left them, and I can pick up working via interface that's 99% indistinguishable, sans the smaller screen size.