Today I re-installed the radio and reconnected the battery after letting it sit overnight on a gut feeling that maybe
it would work (even though I had tried this previously). Scha-zam - the darn thing started working again. It's a mystery that I can't explain, but it works now.
An easy way to see if your factory radio (09 Wrangler) powers up (assuming fuses are good) is to remove the dash as depicted in the video below. After this, pull the plugs from the back and remove radio. There are two bigger sections in the rear of the radio for plugs. They are separated by a smaller plug in the center. Looking at the back of the radio, you will be focusing on the big plug in on the right side.
Inside this are two larger flat pieces. You will need to connect a 12V battery to these pieces. Connect a negative (black cable) to the flat piece on the left, and a positive (red cable) to the piece on the right. To check if power is running through the radio, hit the eject button for a CD. The radio should do something as this point. All the supplies to do this can be picked up at Radio Shack. Cables with clamps at the end are helpful.