"Often" as complement of BE

Question

Often is an adverb according to Oxford Dictionaries Online.

ADVERB

1 Frequently; many times.

'he often goes for long walks by himself'
'how often do you have your hair cut?'

1.1 In many instances. 'vocabulary often reflects social standing'

As an adverb, it should not be a complement of BE as this answer explains and as Huddleston & Pullum (2005:123) say in A Student's Introduction to English Grammar:

Most adjectives can function as predicative complement as well as noun modifier, but adverbs do not normally occur in this function. Again the difference is most easily seen by taking adjective-adverb pairs related by ·ly:

MODIFIER
i. impressive performance [Adjective]
ii She performed impressively [Adverb]

PREDICATIVE COMPLEMENT
i. Her performance was impressive. [Adjective]
ii. *Her performance was impressively.[Adverb]

However, on page 253, Huddleston & Pullum use the following sentence as an example of it-cleft construction.

It isn't often that she misses a class.

This it-cleft sentence derives from its canonical counterpart:

She doesn't often miss a class.

Often in the it-cleft construction is used a complement of BE. So the question is, how can the adverb often function as a complement of BE? Is the 'cleft' construction an exception?

Answer 1

It isn't often that she misses a class.

AdvPs can occasionally serve as predicative complement to the verb "be" in its specifying sense. Here, "often" is complement to "be" and antecedent for the relativised element in the relative clause that she misses a class.

Another decent example is It is now only occasionally that we travel abroad, where the AdvP only occasionally is complement to "be" in its specifying sense, and antecedent for the relativised element in the relative clause that we travel abroad.

And AdvPs can occur as complement to "be" in non-clefts too: The best way to cook it is very slowly, where "very slowly" is complement to "be", again in its specifying sense.