I agree wholeheartedly with the article by Geoff Pullum that you gave a link to. And deadrat is right on the money here too. There's no doubt that although content clauses are often complements, they can’t always be assigned that function, as the examples that Geoff gave in his article clearly demonstrate. His final comment is particularly telling:
"So the property of being a finite complement clause is distinct from the property of being a content clause, despite the fact that nearly all content clauses function as complements".
In any case, clause is a category and complement is a function, so the term 'complement clause' really makes no sense. And worse still, some people even call them noun clauses on the basis of some functional analogy with nouns. The term 'noun clause' is highly misleading. Content clauses aren't nouns; they don’t behave like nouns or NPs; note for example that nouns and NPs don’t normally function as extraposed subject or as complement to a noun or adjective. And consider this example:
“The suggestion that they cheated was quite outrageous”.
The subordinate clause in bold would traditionally be classified as a noun clause, yet it’s not replaceable with a noun. So the analogy fails, and along with the term ‘complement clause', the term 'noun clause' is best avoided.
Incidentally, for the syntactically uninitiated, there are three major subclasses of finite subordinate clause:
Content clause - Relative clause - Comparative clause
The first kind, the content clause lacks the special properties of the other two and is regarded as the default kind of subordinate clause and hence 'content clause' is a very important label indeed.