It won't die when pulling the relay because it has another power path to the fuel pump through the oil pressure switch .
Fuel Pump Electrical Circuits Description and Operation watch this video on fuel pump control , shows chrysler first , ford ,then gm . wait for gm an he explains the oil pressure switch . Plus when it won't start ,don't guess or theorize ! Test what's missing ? check for spark , no spark check ignition system , coil's , ignition module , crank sensor etc... has spark ,do a fuel pressure test etc... don't think computer , when it won't start does the check engine light come on with the key on run not start . If light is lit ,nothing wrong with the cumputer . If want to fix cars you really should take some automotive classes . Test don't guess .Watch some other videos crank no start videos etc...