Very common problem. You have a lot of resistance in your starter wiring. The purple wire does not have enough volts to engauge the solenoid when it gets hot. Your pushbutton works because it bypasses the resistance in the regular factory starter circuit. You can trace down the circuit and find the source of the high resistance or just add a Ford starter solenoid to the circuit and be done with it. Do a search on this, it has been discussed many times.