I suspect you are incorrect in thinking the dealer is trying to get out of making it right. There is no reason for the dealer to to mislead (lie) about the problem. If there is a defect, Goodrich will reimburse him. As someone else asked, “Did you buy Road Hazard protection?” if so, your tire is probably covered.
I agree with whomever said you should have the dealer show you what was in the tire or the hole if the item was no longer there.