Thank you for taking the time to give us this feedback.
Toyota remote start systems are designed this way intentionally. It's for the purpose of vehicle security. The system is actually tied into the door lock detection switch, so it can be started remotely, but as soon as the door is unlocked (with the key or remote) the engine dies and needs to be restarted. It's to prevent someone from being able to steal the vehicle. Unfortunately, there is not a way to diable that feature as Toyota has it programmed into the wiring. I hope that at least gives a clearer explanation as to why this is occurring. Please let us know if you have any additional questions.
Toyota of Puyallup