It would still turn over if it was the CPS and it would probably trigger a check engine light. From what you've described, I would go back to the battery first, make sure the connections are super tight and the battery is good.

I put an electronic ignition kit in my old speedster, going down the highway it just quit. It would turn over great but just wouldn't start. My negative battery cable was just tight enough to let it turn over but not tight enough to give my ignition a good ground. Up until that moment, I thought if it had enough juice to turn the motor over, the cables were fine. It made me a believer on making sure those cables are tight, especially on our electronic enhanced cars.

Dave