On bootup sometimes fsck cannot fix everything it drops the
user into some kind of maint mode and reports to the user
that fsck needs to be run without flags.
fsck should always give a very stromg warning that it is
dangerous to run on a mounted file system if that attempt
is made. fsck called me a probable "bonehead" after it had
already made some changes. I laughed, but wished it had
stated it earlier and informed this "bonehead" how
to fsck the root filesystem without mounting it.
Fortunately no damage was done and I have a sense of humor
about the incident.
fsck'ing a read-only mounted file system (like the root filesystem in the
emergency mode you're referring to) can't hurt.