The videos are out of sequence and if you choose to manually calibrate one esc at a time the red wire must be intact in order to furnish power to the receiver. So yes, you will have to put the red wires back in and calibrate each one.
However, since you have already removed them, it would be simpler to calibrate all four esc’s together. Do this by wiring up the drone in the final configuration, ie, put the esc pwm wires (the wires we are talking about) into the flight controller with all power (red) wires removed save for one in order to power the rail. Then follow the calibration procedure with the transmitter. Using this method you save time in that all are done in one go. You should then observe all motors to spin up together properly.
As a side note, there is a convenient method to observe the behavior of each motor in Mission Planner located at SETUP > Optional Hardware > Motor Test.
For more information see
https://ardupilot.org/copter/docs/esc-calibration.html