iPhone immediately turns off after ~30 seconds of displaying the apple logo

Have you tried doing the same thing but with the cable already plugged in? I had a 3Gs doing the same thing last week, I replaced the battery and that let me put it into recovery mode and then it restored and is now working fine. Possibly your battery is causing your problem. I know you have a 2G iPhone and they are totally different to the 3G and 3Gs, I have 3 2G, 3 3G, 4 3Gs and a 3G iPad - but then that is the business I'm in.

Press and hold the home button as you press the power button, once the "Apple" icon appears release the power button but keep pressing the home button. This will put the phone into recovery mode. Should take about 10 - 20 seconds. You should see the USB cable showing to connect to iTunes on the screen. That is recovery mode, then plug the USB cable in and follow the directions in iTunes to restore. That will restore the phone back to factory settings. You will lose all your data unless you have previously backed the phone up. I hope this is not too hard to understand.

Happens in iPhone 3G and 3Gs also when there is corruption in the phone. Try to restore the same version as in the iPhone at present.