And what is the voltage on the constant wire when the car is off? Still 12v? Is it fused somewhere? because it needs to draw a lot of power, so a blown fuse would mean no go, but if the fuse still had a bit of metal it could send a few mA, and that is all it needs for the remote so maybe that is why it turned on.
That does seem weird. But my guess would be on the 12v constant line. Because the computer powered up fine using the ignition source as the constant source, that leads me to believe it is fine. The 12v constant source is the mystery. Check that closely.