Offering to pay the tax does trigger more tax; but as the added tax is less than what you add to the price, the total amount converges (as long as the tax rate is less than 100%). To have the recipient end up with a net payment N after taxes you've to pay N/(1-r), with r being the tax rate.

With a 25% tax rate and car value of $28,500; to make it a tax-free gift Oprah should've paid an additional $9,500.