This is consistent with no "OK" being sent back from the module to the serial port of the Arduino.
We know that AT mode does not function correctly.
What happens when you connect a phone. What happens with the mode light flashing pattern. Can you turn the led on and off?
This is still really weird because it works really well some of the time
What does this mean? Data mode between the module and the phone is working correctly, or that AT mode sees an OK return sometimes?