I am not a mechanic, so do not treat me seriously, but I have been through a couple of issues with my car and perhaps I can explain as amateur to amateur.
Yes, the role of CSV is to deliver additional fuel during start (not only cold start, but that is what everybody thinks).
Assuming that your engine is properly tuned and ok (compression, valves, ignition), yes the CSV may be the issue. But then it is the CSV itself or a number of other devices that switch it on and off.
You have it all described in the manual under Cold Start Aid. There are many systems across the years, but in essence: when you turn the key to crank the engine you activate the Thermo Time Switch that, if the engine is cold, sends signal through a relay to CSV to open. So if the CSV does not work it canbe CSv faulty, TTS faulty or the relay faulty. From what I read on this forum, the quilty is the TTS, but we do not know it for sure.
One simple test is the one with lamp described under link.
The other simple test I would do would be to disconnect the wire you now have on CSV, put it aside so that it does not touch anything, take a length of wire and connect battery + to CSV connector while cranking (only while cranking and perhaps a second or two longer). See if engine starts better. If it does - need to check TTS and relay. If not - closer look at CSV - does it get fuel, is it clogged (it looks like windshield washer nozzle), does the solenoid work?
