PATS disables the ignition but I don't think it would be intermittent, and light on dash should flash.
The PATS keys have a battery inside and need to be replaced sometimes, so maybe the key you are using is on its way out, but you should get the dash light.
How about this test, you said that after it dies it won't restart unless you turn the key off then back on.
What about getting a can of Quick Start, ether, and the next time it dies don't turn the key off, try to restart once, if it's still dead pop the hood and spray some ether into a larger vacuum line on the intake, i.e. pull off the hose for the PCV or Power booster and put it back on, you could pop the air cleaner but not sure on your setup of the MAF and ether, I doubt it would be an issue.
Then turn over engine(without turning key off), if it doesn't fire then spark is missing not fuel, if it does fire then fuel is missing not spark.
And just as a side note I did find a post about a Saturn that would die and then no start, turned out to be the CKP but only when it got warm, tested fine cold, so I have now heard of an intermittent CKP.
What's the temp like under the hood with that 5.0l in there, just wondering if something has gotten temp sensitive over time?