Yes, do this first it is easy to check. It is easiest if you have someone to hold the test light (oops on my part, not a trouble light) the kind with a metal point to touch to the solenoid and a clamp to go to ground. Then you just try to start the car, if the solenoid is sending power to the CSV the test light will come on. You only get one try or maybe two tests a day, the car needs to be fully cold.
And yes, the car can start - it just takes quite a few cranks to get it to fire off. I have had this happen a couple of times. The first time, I just opened up the cold start relay and filed the points to ensure they were clean. The second time I just replace the relay, it was not that expensive, and kept the old one as a possible back up.