Just to clarify: You measure 5V on servo connector coming from ESC when battery is connected, but USB is not connected and everything is wired normally (ESCs signal/servo connector is connected to FC pins). Correct? If it is lower than say 4.8V then it may be a problem.
If that is the case, it seems there is a rash of this happening lately. I recall talking to at least two other people in the last 2 months that have this problem. Some form of CC3D that works correctly when just USB is plugged in, but not when just flight battery is plugged in.
I believe that the first person to have this issue finally just connected from the USB port to the ESC/servo connectors some way. So when power comes in the servo connectors it is forked to the USB where it can actually get in.
To help debug this problem, you could connect 6V to the FC as if 6V was coming from the ESCs and see if it starts to work then.