If you jump out starter relay pins #30 & #87 and starter motor cranks the engine, then most likely you have a bad starter relay or it is not getting a signal from the ign. switch via the ECM to engage the starter relay.
Try swapping out the starter relay with one of the other same size relays like the wiper or the viscous heater relay and see if it cures your issue.
If it does, replace the relay with a new one.
Also check all wires at starter to make sure all of them are good and tight with no loose connections.
Starter Motor Relay Terminals
Terminal #86 -- supplied 12v by the Ign switch from Fuse #28 (15A) in the TIPM under the hood
Terminal #85 -- supplied a *ground by the ECM (engine control module) if all conditions are satisfied.
*(only when ign. switch is turned to the "START" position)
Terminal #30 – supplied 12v directly from the battery via a 40A fuse #8 in the TIPM under the hood
Terminal #87 – supplies 12v directly to the starter motor solenoid when the relay is energized
Terminal #87A -- not used