So the car won't engage the starter on it's own?
And the car won't run IF you get it to turn over as you described?

THen check to see if the output of hte Engine Control Relay (on the RCM) is getting to the PTCM.
That output not only "allows" the PTCM to operate the plugs and injectors, it also is required for the PTCM to command the pulse module to engage the starter.