Ran into the same thing. What they don't tell you is there is no way to retain power to the computer when you crank. You need to hook up a push button starter switch to 12V hot and the start input to the Hotwire harness. It should fire right up.
To start turn the ignition switch to Run and then push the starter button.

This happened to me 2 years ago, it's possible they made some change to their harness.

This was in a 73 Chally, we hid the push button start in the headlamp/wiper switch bank. Drilled out the blank one and put a simple push button from Napa in there. It looked factory. I'll dig up a pic. We also put electric truck popper in the same switch bank.