If the radio component works, then I'm guessing the shuttle mechansim that swaps the disks is jammed. My only suggestion would be to completely power cycle the unit, which is most easily done by removing the negetive lead or ground strap off the battery under the hood. Wait a few minutes and then hook the strap back up. Many decks will automatically eject the disks inside when power is restored, but that varies from model to model. If this doesn't work, well you've already considered the alternatives. If you download and follow the troubleshooting procedure from the service, it basically says "replace CD player".
If the radio component works, then I'm guessing the shuttle mechansim that swaps the disks is jammed.
When I made this statement, I was assuming that fuses were not an issue, but that may not be true. Hyundai's are known to have multiple fuses protecting the audio systems. Check the fuse box under the dash to see if any of those are blown.