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More likely it is the brake light switch that is activated by the brake pedal. Either it is defective, out of adjustment (brake pedal is not activating it), or the wiring harness plug worked off it (unlikely). Unplug the connector on the brake light switch and jump the connections on the connector and observe if the brake lights come on. There are two sets of contacts in the brake light switch, one for the 4 lower brake lights and one for the high (or middle) brake light. When unplugged, you can use your ohmeter to test the brake light switch.
The switch usually doesn't go bad, but the bracket that holds the stop light switch is an "el cheapo" plastic part (used to be metal, but GM probably figured they could save 2 pennies) that cracks and allows the brake light switch and/or the cruise control vacuum cut-off to go out of adjustment. It's ~$7 at the dealer, part #3537079.
You'll have to remove the DS knee bolster, lay on your back and get your head under there to verify and replace this part.