If you pay the tax a person owes for a gift you gave someone...isn’t that tax paying amount still considered a gift? I don’t see how Oprah paying the tax for people she gave a car, or a trip to, can be right.
Not to wade too far into the tax-code weeds, but it depends on what 1099 the "giver" sends to the IRS. The tax on gifts (Gift Tax) is generally paid by the donor. IOW, if you receive a gift from your wildly rich uncle of $1M, then the uncle also owes (roughly) $500K in US federal gift tax. The gift is reported on the uncle's tax return.
Now, Oprah has a choice. She can report the trip to her audience as a "gift" - where she is responsible for the tax, or as a "prize" - this is where the recipient would be liable for the tax. It would be then recorded on the recipient's tax return, and the market value of the prize would be counted as "income".