Here is a description of the choke system. If the choke is not getting power you need to check the circuit from the alternator to the choke.
On this system, a special tap is provided at the alternator to supply a 7 volt power supply to the electric assist choke. A thermostatic switch within the choke cover closes when the interior of the choke cover reaches modulating temperature. This allows current to flow to the heating element when the circuit is completed through the choke cover ground strap and choke housing to the engine. As the heating element becomes hot, heat is absorbed by an attached metal plate which heats the choke bi-metal coil. When the engine is turned off, the thermostatic switch remains closed until the choke cover interior modulating temperature is reached. If the engine is started immediately after shutdown and before the modulating temperature is reached, current will immediately begin warming up the heating element to open the choke. Once the choke cover interior temperature falls below the modulating temperature, the thermostatic switch opens and current flow to the heating element is shut off. If the engine is restarted at this time, the electric assist function will not operate until choke cover interior reaches its modulating temperature.
If the choke is good, tapping into the alternator stator wire to power the choke will make things operate fairly well without all the tail-chasing needed to trace the original circuits.
are you looking to save $$ or get the jeep back to OEM status? thats the question. you need to figure out the answer you want to spend.
after reading the replies i am reminded of the bi-metallic strip mentioned. they wear out after a while. since your jeep is a few years old i would not be surprised that it is worn out.
these parts ARE readily available, but you need to replace two or three items to get it back to work as it used to. then it is susceptible to breaking again. a manual choke is pretty dumb proof. and once you get used to it, and remember to turn it off when its warmed up they are simple.
a manual choke cable costs maybe $30 and hooks to the existing choke plate, thus eliminating the old stuff that doesn’t work correctly.
a vehicle of your age is on the edge of the era when cars had mechanical chokes, operated by sensors, and mechanical levers, and the later era of sensors, relays and solenoids. if you had a slightly newer car (with computer, and electronics) you would not be able to easily convert to a manual choke, but since your car is able to easily have a manual retrofitted, at least that is an option.
Pretty smart of me to forget the link. The link provided by Skypilot is the one I forgot. Note the 4 cylinder uses a choke relay instead of a switch and the alternator tap is used to kick the relay. The 6 does not tap into the alt.