Well one of the symptoms of a bad CPS is that it will start up, run for a bit (time varies) then just die. Other symptoms include a loss of the traction control system, no tach signal on the dash.
If the CPS were completely dead and fried, then no, it probably wouldnt start because the Ignition control module would not be getting the signal it needs to fire the ignition coils. However, I have yet to see one just go flat dead. They usually become intermittent first.
One way to test the CPS is to start the car up, let it run until it dies, then test for spark at the #1 cylinder. If you have no spark the most likely culprit is the CPS.
Now, all that said, I have a problem with the fact you found black cloudy gas out of the fuel filter. This could be an issue as its possible that the filter was severely loaded with gunk and if that is the case it is likely some of that slipped through and perhaps clogged up one or more fuel injectors. As well, just because the car runs at all, does not mean that the FPR is good.
What you need to do there, is get a fuel pressure tester and see what kind of fuel pressure you have when the car is running.
So what you need to do now is try what I mentioned and report back what the results are. We'll have to rule things out that are likely to be causing the issue before we can narrow it down.
