The failure in the RCM most often takes the form of no output to the ECU. This means the car seems to power up, but the starter will not engage.
If you leave the key at "ON" for about 45 seconds, the engine fan will warp up to high speed. This is a clear sign you have no output from the RCM to the ECU.

But you are saying that the starter is now engaging and the car still won't start. The only failure on the RCM that can cause this, is a failure of the fuel pump relay - that does happen, but it's rare.
YOu should hear the fuel pump run for a few seconds when you turn the key to "ON". You can verify you have fuel pressure with a cheap fuel pressure test kit ($30-60).

It is common for cars to quit running (but the starter engages) with a bad Crank Position Sensor. USUALLY, they fail at high temps, letting the car sit and cool will usually mean the car will start and run until it gets hot again. Perhaps your crank sensor IS bad, but with a cold engine, I'd almost doubt it.

YOu need ignition and fuel and air for the car to start. Air is the easy part, what is questionable is if you have ignition and fuel.