I BELIEVE that the lightning connector doesn't put out raw audio. Only a digital form that needs to be decoded. That's why a lot of the car connectors come with the lightning and 3.5mm mini jack
as for the 3.5mm mini jack, as for every accessory in the ford's they are sh*t cheap quality. They would have tiny little soldered connections and I bet you would have a bad or failing join on one of em. Best things to rip it out and see if you can see anything loose or just replace the whole circuit. Test for continuity with a multimeter if you can before replacing it