You need a wiring adapter that specifically works with the factory amp, not the one that you purchased. The one you linked specifically says it will not work with the factory amp.

Try this one instead: https://www.amazon.com/ASC-Premium-Harness-Chrysler-Vehicles/dp/B079R75FRV/ref=sr\_1\_15?crid=1EF7IE4EGKA74&keywords=jeep+patriot+can+bus+adapter+radio+harness&qid=1685928507&s=automotive&sprefix=jeep+patriot+can+bus+adapter+radio+harness%2Cautomotive%2C102&sr=1-15&ufe=app\_do%3Aamzn1.fos.006c50ae-5d4c-4777-9bc0-4513d670b6bc