I might have to rotate the Chevelle into the mix more often if this car doesn't quit acting up.

2001 Malibu 3.1L 113.5K miles.

I keep up with regular maintenance pretty well, it gets regular oil changes and has been running good synthetics (Mobil 1/Amsoil) for a long time now. Hoses, filters and belts get replaced when they need to be. Recently one of the four O2 sensors was acting up, as was the thermostat but that's expected with the mileage and easy enough to diagnose/fix.

Cooling system was flushed at around 70k miles.
Fuel system was thoroughly flushed, tank to injectors - 2 summers ago.

This has been a VERY reliable car but it's doing something weird recently. The engine will crank over, run for maybe a second and then die. I'll try again and it'll die again. But if I push the gas pedal down about 1/4" and then crank it, it'll stay on. Sometimes I can let my foot out right away and it'll stay on, other times it'll want the throttle open for a while before it'll run on its own.

When it runs on its own right away, I don't get any idiot lights going on. When I have to keep it revved for a bit, the BRAKE and ABS lights will come on and stay on, but the brakes work just fine and the lights will go away if i shut the car off and start it again. Once the car starts it runs flawlessly and the throttle response feels as good as ever.

Any ideas?

Depending on what the injectors were flushed with, it may have destroyed them. Possibly a bad fuel pressure regulator as well.

Depending on what the injectors were flushed with, it may have destroyed them. Possibly a bad fuel pressure regulator as well.

The fuel system was flushed by a good local shop. I would think that if the injectors were ruined by the flush they would've started acting up right away? It's been over 2 years (and over 20k miles) now and this problem just started recently.

Could you elaborate on how the Fuel Pressure Regulator would be helped by revving engine for a few seconds at start-up?

Fuel pressure regulators are controlled by engine vacuum, when they go bad, you lose fuel into the engine through the throttle body. Fuel can puddle in the intake runners and cause a rich condition, which is cleaned out by opening the throttle.
It can be many things that cause this, but without the car in front of me to see what it does, it is hard to say what it could be.
If it's been a couple years since the flush then it was not the flush that has hurt the injectors. It still can be the injectors, it is not that uncommon to have one or two that might be getting bad, and leaking fuel, causing the same problems. If you have to crack the throttle to get it to start, that tells me that most likely there is a rich, or overfueled condition. Might have your shop run an injector balance test.

Have someone scan it when cold to see if the temp sensor is telling the computer that the engine is up to temp, that can cause what you are getting, Also if the IAC (idle air control) is stuck or balky, it might cause your problem.

sounds like fuel pump and/or pressure reulator. I've got one of those sitting in my driveway right now.