John, if you let the key stay on for a few seconds and then start without idle issues then I think you have found your problem. When you turn the ignition the fuel pump starts up and pressures the system if you start straight away it does not get the chance to pressure the system enough. I would agree with you on it not being the cold start system now. Another way to pressure on old cars is to turn the ignition a couple of times as the fuel pump only runs for a predefined number of seconds.
I would look into why you are losing the pressure. Which is where you are already. Sorry was not much help