Back in November, in post #6 in this thread I suggested that the fuel p[ump was not being powered during a R/S.
I was right. I know these cars. I'm quite familiar with 1996 Riviera, as I maintained one for a neighbor for 10 years. Just for the record I'm an ASE Master, certified in all mechanical areas. I've been wrenching cars, boats and yachts for almost 50 years. I am VERY familiar with GM wiring and engine strategies.
Very first thing:
Does it start and run normally with the key?
If = YES it is your connections from the R/S to the IGN wiring that are at fault. It is NOT the car. There is no single wire from the IGN switch for the fuel pump. The pink and white wires do it all.
I also see conflicting reports. You say "fuel pump not powered during crank". But the initial problem was "starts, but dies-out [from lack of fuel] after it quits cranking". These are two different sets of symptoms, with two different causes. Which is it? In the first, it shouldn't even start, because the pump was never on. In the second, the pump was on during cranking, but turned off.
There are three IGN wires. Orange, Pink and White.( I do not see a "pur/wht" wire in the diagrams at the IGN switch. In an earlier post, I explained what pur/wht in a GM is. It is NOT to be powered by a R/S.) Orange is important because it powers the electric cooling fans, without it, the car can overheat during a R/S because the fans won't come on. If pink is not powered, it will not run. so your pink is fine.
If white is not powered, there is no request to the PCM to fuel the engine. The PCM controls the fuel pump.
Check the white wire again.
If it is 12V in run with R/S,
but not during a R/S crank,
this is your problem. You probably have the white wire configured as an ACCY output rather than an IGN.
If the white wire is not powered-up during CRANK, the fuel pump will not be enabled.
Take off all the band-aids and work-arounds. Figure out what the real problem is. Probably that you have the output feeding the white wire programmed as an ACCY, and it needs to be IGN.