When I connect to the system for calibration of motor direction all the motors were rotating without any issue.
In this case the Fc is powered correctly trough USB port.
Seems every time you apply the Throttle, the FC reboots.
Usually we assume the power wires will have almost the same size as the ones used on battery, but in your build wires are really small.
How do you power the Fc with +5V ? The small module with yellow capacitor ?
Double check voltage and add ground wire between module and Fc. (looks like there is only one wire)