It is my understanding that fuel enrichment solenoid has very little effect if anything when the engine is cold. I know for a fact it doesn't on my car.
The FES is always activated when the starter is engaged, hot or cold engine. But depending on the FIP rack position at the time (forward when engine cold, back when engine warm) the solenoid pin may not move the FIP rack.
An easy test to see if the solenoid is moving the rack when the starter is activated is screw in a ~3" long 5mm(?) bolt in rack port in the front of the FIP . There usually is a rubber cap that needs to be taken off or earlier pumps have a cap that needs to be unscrewed. I can't remember exactly.
Place a mark on the bolt with a sharpie pen and note it's location relative to the pump.
Now when engine cold activate solenoid and see if FIP test bolt moved. I bet it doesn't. That's normal on my car.
When engine is warm/hot you will notice the bolt and the "mark" have moved back into the pump. That's because the WRD (if working properly) has leaned out the pump and move the rack back to its warm position. These are only small movements.
Now, when engine warm/hot activate the FES. I bet you will see your test bolt move forward. That means the FES pin was able to reach and affect the rack thus enriching warm/hot starts. I believe this to be normal.
Long story short. I don't think this is your cold start problem.