I think you were close to the mark when you said that the dwarves were not made by Illuvatar. Although it was Eru who allowed the dwarves to live,(which was a gift in itself, as you stated) it was Aule that created them, and so its probably not in his power to give the type of gift that Illuvatar could give. But, there are much deeper interpretations and ideas concerning this, this is just my surface opinion.
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