Engine is hard cold start?

Son's truck. Symptoms: Hard to start when even mildly cold; rough idle always, low power, dies when coasting or gearing down to stop, poor mileage.

History: Previous owner had engine rebuilt 8,000-10,000 miles ago; ran good. Buoght it, gave it to my son (who is pretty hard on vehicles) and problems began within a month. Good history with previous owner (50+ y/o, 99% highway user, took good care of vehicle).

What I know: Two immediate problems I see. The coolant temp sensor is broken (yes broken). The inside comes out with the connector and won't stay in place. Not sure if it actually works but I will be replaceing it when next I have access to the truck. The thermal time sensor connector also wont stay attached. Again, not sure if that's a factor (maybe for start) because it shouldn't play a role once the engine is running (right?).

My plan is (to troubleshoot the cold start issue) to connect a push to close switch diectly to the cold start valve from the battery and to ground (+ bat to terminal 1 CSW through push to close switch and ground wire directly to the terminal 2 of CSW), whereby bypassing the thermal time sensor completely. Pushing the button while cranking the engine (in my little mind) should tell me if the cold start issue is in either the thermal time sensor or the coolant temp sensor. Right? If the problem continues, it must be somewhere else. Right? More to follow on that if successful.

So, to the question: Should any of this affect the overall performance of the engine once it is started? And if not, where do I start?

Thanks in advance

Lowflyer