Although it's true most native speakers would reply "I left it at home" in this exact context, that doesn't imply "I have left it at home" (or contracted I've) is somehow incorrect or otherwise inappropriate.
The Present Perfect is normally used when you want to make a connection between the past action and the present moment. But in OP's example there's no need for the more complex tense; the connection between past/present is semantically implicit anyway, so you may as well use the simpler tense.
There aren't actually many contexts where the choice between present perfect/simple past makes much difference. One that comes to mind is...
1: He earned my trust [in the past, so I lent him some money, which unfortunately he never repaid]
2: He has earned my trust [so I am now going to lend him some money, which I'm sure he will repay]
...in those examples, the bracketed supplementary text only really works with the tenses as given.