Think for a moment about what happens within the EFI system and the engine in order to get the engine running and this is true for both the turbo 22RTE(C) and the 22RE(C)....
The ECU measures all sorts of things including the engine temperature (determined by the coolant temp sensor), engine rpms (determined by the distributor pick-up coil and igniter module)... how far the throttle is opened (the throttle position sensor), how much air is entering the engine (determined by the air-flow meter on top of the air filter) and how 'warm' (and consequently how dense) that air is. When the engine's warmed up it uses the O2 sensor to fine tune the fuel mixture.
To simplify things a bit, the ECU programmers figured out that at a certain RPM the engine needs 'X' amount of fuel and timing. They programmed in 'Maps' that the ECU could read in order to get the amount of fuel required and then the ECU uses the sensors to adjust that amount according to temperature, throttle opening and feedback from the O2. Had they not done that, the ECU would have to sample all those sensors, do the math and calculate how much fuel should be injected and that would require a lot of CPU time (consider we're talking about a system from the 80's) in order to calculate that on the fly.
Now... you're having an issue with the engine dying after starting. The fact the engine starts rules out
most everything in the fuel and ignition system- apparently the injectors are working and the spark plugs are firing. So what would cause the engine to die after a few seconds of seemingly perfect operation? One has to consider exactly
how the EFI system works on our trucks in order to get to that answer.
When you turn the ignition key and start the engine, a few things happen. The obvious first thing is the starter spins and cranks the engine over- necessary to get the engine running. The second thing is the Circuit Opening Relay (COR) is energized thus supplying voltage to the fuel pump in order to deliver fuel to the Cold Start Injector and the fuel injectors in general. Given that your engine starts and runs for a few seconds implies both of those systems are working at least somewhat as designed.
We end up with a problem though (if you think about it for a moment) when you release the key after the engine starts- the COR stops operating and would cut power to the fuel pump causing fuel to stop circulating and the engine to die. Toyota figured that out.
They put another set of contacts in the air-flow meter on top of the air filter which turns the COR on (and thus the fuel pump) when there is enough air flowing through the meter to keep the engine running. So in a nutshell: the ignition key turns the fuel pump on and cranks the engine over to start it and the air-flow meter keeps the fuel pump turned on once the engine has started running.
Logically speaking, the fact the engine runs for a few seconds then stops running implies that the fuel is being cut or that there is too much air getting into the system around the air-flow meter. So a large vacuum leak, like one would have if the main pipe to the brake booster were leaking, could cause that. A crack in the intake duct from the throttle body to the air flow meter could cause that as well. In the first case, the fuel mixture is too lean to keep the engine running, and in the second case the air-flow meter is not being opened enough to turn the COR on.