1988 silverado, tbi 350 has 12 psi fuel pressure, replaced oil pressure switch and fuel pump relay. starts and runs 2-3 seconds and then shuts off? injectors are fireing good at start up then quit.

4.3L , L35(RPO engine)
has 1 fuel pump relay PCM controlled
has 1 fuel pump.oil pressure sensor , too.
not sure at all me what a pulsator might be.Ahh Dampener.
what matters is spark first, it must have spark or the PCM cuts fuel dead on all EFI car. (if the PCM detects spark bad in ANY WAY)
the comes cranking the pump just receiver 12vdc cranking
KEY ON ONLY is only 2sec prime puls best is forget that unless have 2 person to test prime, but cranking matters first.
some new pumps are silent, (like top Bosch) so sounds prove little. today what really matters is PSI pressure cranked,
the PCM brain or the oil pressure switch allow the pump to have power cranked, and running.
that is how to test it, cranked only.

"starting with start fluid,but now it just cranks over ,only."
does this mean starting fluid now fails.?
we check spark at all spark plugs no just 1.
we check oil pressure to, if zero PSI cranked that is bad .
last is fuel pressure if not at spec, it will not start.
spec fuel pressure is 58-64 psi
yes some fuel rails or near by or near tank, have pulsator that is really regulator/dampener.
it prevents old cheap pumps from madly pulsing the rail and causing injectors to mess up. vane pumps do make pulses some worst tan others, (some even by pass it, but there is their call)
Is this a TBI throttle body injection? or MPI.

the SM states this. (it has a 2 second prime for sure)

If the engine does not run, turn the ignition ON, but do not attempt to start the engine. Listen for the fuel pump to run. Within 2 seconds of turning the ignition ON pressure should be 58-64 psi (400-440 kPa) while the pump is running. Once the pump stops, pressure may vary by several pounds, then it should hold steady. If the pressure does not hold steady, wait 10 seconds and repeat this...

Fuel pump relay may be weak. Try removing the connector from the oil pressure switch and connecting the terminals together so as to bypass the switch.
if this works, replace the relay.

Oil pressure switch puts power to the pump after it is started.

Pumps also tend to loose their ground with age. I often splice in a new ground at the pump to eliminate that problem area.

Need some votes, thanks

replace the oil sending switch on the motor. upon starting the engine this switch has a 3rd wire connected to the fuel pump
when oil pressure is running at its normarl psi the computer will shut that circuit off and then turn on the fuel pump relay

The fuelpump is primed by the oil pressure switch.When the fuel pressure reaches normal PSI the computer shuts that pressure switch off and turns thepoweron to the fuelpump relay.You have a bad fuel pump relay

without starting the engine position the key to the on position.
you can hear the fuel pump operationing for 4 to 6 seconds upon having the key on.
if not heard running check the fuel pump fuse, the fuel pump relay, and the engines oil pressure switch.
they all work the pump you see
the engine oil pressure switch runs the pump untill the engine reaches normal pressure
once the pressure is reached to the normal range the computer shuts that circuit off and turns the fuel pump relay on

turn the ignition to the on position without starting the engine. you should hear the fuel pump running for 5 seconds inside the fuel tank. if not heard replace the engines oil switch. that starts the pumps circuit when the key is turned on. after the engine builds up its normal oil pressure the computer shuts that oil switches circuit off and turns on the fuel pump relay

You need a new fuel pump relay. The engine fires up and runs off the oil pressure switch untill the oil pressure PSI is reached at normal operating conditions. When the oil pressure reaches its operating PSI the PCM or engines computer will shut off that circuit and turns on the fuel pump relay to run the fuel pump untill you shut the engine off

it could be a clogged or failing fuel pressure regulator

On that one it actually called circuit opening relay under left side of dash in fuse relay box