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.