An enterprising new owner!
Well, the cold start system has basically, as you've learned, an
electric cold start valve, a temperature sensor and a time-temperature
relay. You already know about the clod start valve so there's no more to
say about it. Rather than powering the valve I'd put a voltmeter on its
lead to see if it receives power during cranking. It probably doesn't.
That leads back to the thermo-time relay which powers the cold start
valve for shorter or longer periods, depending on the engine's
temperature. We're talking seconds, not minutes.
Finally, there's a temperature switch, ON below XX degrees, OFF above that.
These three parts are in a chain, failure of one kills the whole cold
start system so your task is to diagnose which is defective and replace
it. My $$ would be on the thermo-time relay.
Hope you solve the problem.