The starter circuit is separate from the asd relay. The asd turns on the fuel pump and supplies power to the 02 sensor coil and fuel injectors. So of course with out it it's not going to run. Did you check if you have good grounds on the engine and body? Also check the connections on the battery.

 

I understand that it won't run without it, but what is happening such that a connected ASD prevents the starter from operating, despite it receiving power as evidenced by the solenoid engaging?