Logically, the radio is plugged in to 12v and giving 5v at the USB socket, so power conversion is happening inside the radio for the USB slot. That should rule out the seat, the cigarette socket isn't linked to the radio in any meaningful way. You've proved the lead and more importantly the USB stick with MP3's doesn't register as plugged in at the radio so it being an Apple thing is a red herring too. If the USB stick isn't getting power but the radio is that suggests the problem is in the socket, or the circuitry to the socket.
If you're comfortable taking the radio apart then go for it, if it's a pay someone job then another radio would probably be cheaper and a different power solution cheaper still (although you'll lose all the functionality at th radio.
Have you searched the model for known faults on the interweb?