quadcopter F450, bumped start. Damaged one propeller.

   esc_1 = throttle - pid_output_pitch
   esc_2 = throttle + pid_output_pitch
   esc_3 = throttle - pid_output_pitch
   esc_4 = throttle - pid_output_pitch

That's part of your problem. To control pitch, two motors should be biased "+" and two motors "-". You have one "+" and three "-".