My thought's are that your rail pressure defaults to the maximum, 24,000PSI, when disconnected (they probably program it like this so the vehicle is not disabled all together in the event of rail sensor failure). I'm surprised it doesn't go into limp mode after starting. Rail pressure needs to be 2,000PSI before the engine will start - so you've worked out how to solve that problem. You could have a faulty sensor, but the fact that it works at all, when connected, suggests that this is unlikely - sensors generally fail completely or intermittently, but rarely fail in a manner to give lower readings. But that's not to say its not possible. Maybe try starting the engine with it disconnected, then connect it when running and see what the OBD2 PID reads.
To me, unfortunately, it sounds like your pump can't produce the pressure required for starting, 2,000PSI. In that case, you will need a new pump. Very expensive.