Jumping the starter relay out gives you good starting you say.
This jumpering of pin 30 to 87 bypasses the ignition switch, actuator pin assembly, Park sensor, Neutral sensor. Basically feeds +12 volts direct from Fuse#8 40 Amp to the starter solenoid. If the chipped key is in the ignition slot and ignition is ON then the engine should start and run.
So you must have a good +12 volts going to pin 86 of the starter relay (coil) via Fuse #28 15 Amp. Make sure that fuse is good and in clean and tight.
Then pin #85 of the starter coil is supposed to get Ground via the ECM Pin #58 of connector C2. This Ground will only appear if the ECM detects that the Jeep is in Park or Neutral.
So if there is +12 volts on pin 86 of starter relay...you need to keep your meter lead on pin 86 and move the other lead to pin 85 to check for ground. If ground is not there you need to move the lead down through connector C111 pin# 10 to check for Ground and then to the ECM pin 58 of C2 connector.
If Ground is not at ECM pin 58 then the Park and/or the Neutral interlocks are not working...try start it in Park.