So your car will only crank by jumping the starter relay pins? If I were you with a 2.2 with high miles, I would want to check compression dry then wet. Verify 180 psi in all for cylinders before further testing. Please report back and I will continue to assist.

 

You need four components for an engine to start and run. Air, fuel, spark and compression. Check for any blockage in the air intake, from the box to the throttle body. Check for spark, check for fuel and, like Andy said, check for compression. If you don't have any one of the four, you need to research why and try to rectify that problem. 360,000 miles is a lot of miles. Have you ever changed the timing chain and components?

 

I made a new post for my issue

Here

Did this on mobile. Will check back when I get home.