Sounds like something in the start chain is not latching. Not sure if this 3.7 Roadster start system schematic helps although it may if your car had the option of start button to trigger start sequence, part of auto stop/start on other applications.
![[Linked Image]](https://i.imgur.com/kTwbh9o.png)
Note the BCU is not involved with key start. Also the PCM (ECU) controls both legs of start relay coil.