It might be that the proof in the paper you refer to did not prove the whole claim, but proved something weaker but still nontrivial. (Imagine for example, a statement of the Weierstrass extreme value theorem without assuming that the domain is not empty. Probably the proof covers the non empty domain case.) In that case, I could try to state a theorem with the additional assumption and mention in a remark or in a footnote that the statement is weaker than the one in the source and that the assumption I have added was needed at some point in the proof I precise.
By the way, when I find a counterexample to a published result, I find it a good practice to identify where the published proof is failing. (Most of the time, proofs fail in the parts left to the reader, but you can still find a counterexample to an intermediate statement in a proof.)