It's just plain wrong to say that there are more facts in a theory than proofs of facts. Both of these sets are countably infinite. Godel's theorem proves that provable facts are a strict subset of all facts.
While I agree with you, in the context of a "hand wavey" explanation, "more" is not rigorously defined and so saying "plain wrong" is silly. A strict subset relationship is a reasonable interpretation of "more" in the context of an informal discussion.
Leaving aside the fact that talking "informally" about infinities easily leads to wrong conclusions, the OP wasn't even talking about a strict subset relationship. The comparison was between a set of proofs and a set of facts, which are sets of different mathematical objects. So at best you can say that each proof proves a fact (a fact can have several proofs) and the set of facts that have a proof is a strict subset of all facts. Or, informally, there are more facts than facts which can be proved. Oh, wait, this just restates the theorem we're trying to prove in the first place and doesn't give any insight into why it's correct.
And if you read about Cantor you will find he defined a term called 'countably infinite'. No 'countably infinite' set has a higher cardinality than any other.
As if there can only be one. (And it seems OP is describing the theorem, not a proof)