Usage of "that" (not as a clause)

I get the difference between that and which in restrictive and non-restrictive clauses, but my question has to do with the usage of “that” in another capacity. I have a friend who insistents that the word “that” is over used in sentences. Case in point: My only concern is that I want you to be happy. He says should read- My only concern is I want you to be happy. To me the latter sounds awkward and perhaps could sound OK if you put a comma after after “concern” or “is”. I still think the first sentence sounds better when you think of how it would sound when spoken. The second sentence also seems to feel as if there was supposed to be some thought or concept in between “is” and “I”. Am I right or is he?

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My only concern is (that) I want you to be happy.- ‘that’ is optional.
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