(in a discussion of "clauses beginning with 'that'")
>Have I missed something?

Yup.  There are two kinds of clauses in English that are introduced
with "that".  One kind is a noun clause (called a "complement"),
which may appear (like a noun) as subject or direct object.  These
are tensed (finite) sentences with a "that" in front of them:
   That you were shocked is perhaps not surprising.
   I told him that you were shocked.
The subject use is somewhat rare because English speakers prefer
to put the verb early in the sentence, and "heavy" subjects like
this are often extraposed to the end and replaced with a dummy "it":
   It is perhaps not surprising that you were shocked.

The "that" may be deleted, and usually is, except when the clause is
the subject and is not extraposed.  In that position, "that" marks
the clause as subordinate and prevents the listener from interpreting
it as the main clause:
   *You were shocked is not surprising.
    I told him you were shocked.
    It's not surprising you were shocked.

That one is the "substantival" variety, because it acts like a
substantive (noun).  The other one is the restrictive relative
clause, which is, as you noted, an adjectival usage, not a substantival
one.

These clauses modify and contain a reference to some noun phrase,
which is then represented in the clause by a "relative pronoun" like
"which" or "who", depending on its animacy:
   The man who(m) you fired just walked in the door with a gun.
   The gun which he is carrying is very big.

In a restrictive relative clause, these pronouns may be replaced by "that":
   The man that you fired just walked in the door with a gun.
   The gun that he is carrying is very big.
Or they may be deleted altogether, except when they are the subject:
   The man you fired just walked in the door with a gun.
   The gun he is carrying is very big.
   The man who/that fired him is in deep trouble.
  *The man fired him is in deep trouble.

Hope this helps.  Incidentally, neither of these is the same as the 
demonstrative "that" of "this and that".

  --------------------------------------------
 -John Lawler                     More grammar
  Linguistics Program   University of Michigan
 "..and, who knows? Maybe the horse will sing."