At last you may be getting somewhere. For my money (which isn't worth so much these days!) I would still plump for an electrical issue (Open earth?). This may be simply down to a bad harness connector, but they are really hard to track down. Until you can be assured you have good connections, please don't throw more parts at the vehicle .....
There are numerous posts (even on this forum) regarding the P codes you have. Especially P1631, which is an odd one. I presume you are in USA??? Then your 2004 should not have an immobiliser which is one problem area for that code. Therefore, something else would be causing a FAIL signal to the ECM (The fact codes are still there indicates the ECM is working) and it is retaining the FAIl signal. Start (fuel/ignition/injection) will remain deactivated until the fault is corrected. ie you will not get the signals you are expecting because of the FAULT, not because the control boxes are not working.
If you are searching on Google try explicitly mentioning Kia (eg "OBDII code P1631 - Kia Sorento" ) to narrow down the results to your vehicle type since each manfr will have there own take on the code. (It even varies between countries - even our old 3.5V6 models had immobilisers and diesels have entirely different systems.)