The following sentences are from the BBC [just to give a source showing that these sentences are all grammatically correct].
- It was Rob that/who ate my biscuits. (not Catherine)
- It was my biscuits that/which Rob ate. (not my sandwiches)
- Was it Rob that/who ate my biscuits?
The list could goes on. My point is, these relative clauses are not defining relative clauses.
In a defining relative clause, we have an antecedent and a relative clause where the relative clause helps the listener identify who or what the antecedent refers to:
- That's the man who ate my biscuits.
- Look for the elephant who's carrying the umbrella.
- The doctor that treated me was a woman.
In (4) the phrase who ate my biscuits tells us which man we are talking about. The clause who's carrying an umbrella identifies which elephant to look for. The clause that treated me helps identify which doctor was a woman. And we see also that in some sense the relative clause restricts our interpretation of the antecedent. For these reasons, such clauses are also sometimes referred to as 'restrictive'.
Because of the way these work, we cannot use such relative clauses when the antecedent is already known or identifiable. For example (7) below only works if you have more than one father:
(7) Yesterday, I met your father that bought me a drink.
I've been told in comments that grammars like CGEL, which recognise that these types of clause in (4-6) are normally defining/restrictive, call them 'integrated' relative clauses. They note that integrated relatives form a constituent with antecedent noun phrases, and are tightly integrated into the construction they appear in.
However, as @BillJ and others inform me, the relative clauses in (1-3) do not form a constituent with their antecedent.
Back to the thrust of my question. In (1-3), the relative clauses do not explain or define who Rob is. The listener already knows who Rob is. So unlike the defining relative clauses in (4-6),they do not restrict our understanding of the reference of the noun phrase Rob. And as shown in (7) unlike with defining/restrictive/integrated relative clauses we can use the relative clauses in (1-3) with a name or proper noun whose identity is already established.
As I understand it, it's usually ungrammatical to use 'that' in a non-defining relative clause. Given that the relative clauses in (1-3) are not defining, are not restrictive, and do not form a consituent with their antecedent, why is using that not a problem when the relative clause is in a cleft sentence?
Also aren't these relative clauses providing extra information?+
+In my original question I assumed that these relative clauses in (1-3) provide additional information. However, as pointed out in comments and in @DW256's answer, they aren't.