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You don't have to start the engine, you can do the same procedure from accessory mode (holding down the start button WITHOUT your foot on the brake pedal (basically key position 2)).
On certain years/models you have to start the vehicle. On this 2018, you must start the vehicle (I tried in run as well and it didn't work). Some of GMs newer years will allow you to do this process with the run like you stated (I also mentioned at the end of the video that some only work with the car on and others will work in run).
No, you do not need to keep the car running. After vehicle is in service mode, shut vehicle off and complete brake job. Then when done turn car back on, the light will be flashing and then do the process to turn off service mode. Sorry I should have mentioned this in the video.