***** A NON-TEACHER's COMMENT *****
(1) As you know, an adjective does not always come before
a noun: The United States is large.
(2) I think that "Nice to meet you" is just a short way to
say "It is/was nice to meet you."
(a) That is:
It (to meet you) is/was nice.
(i) In such a sentence, the adjective "nice" refers to the subject which is
"it." It is/was nice. "To meet you" explains the "it." (It is nice; what is
nice? To meet you.)
P.S. I think that your students would be amused to learn that
many, many, many years ago, "nice" = stupid or ignorant!!! Just check
any good dictionary about the history of that word.
Student or Learner