What prevents the other original brake light from blinking?
The other brake light is on a different wire. An 1157 bulb, which is all E body tail lights, has a running light filament and a turn/brake filament. The wire supplying the brake/turn element in the bulb is energized on both left and right when the brakes are applied. When a turn signal is activated, it only energizes the appropriate side. However, turn signals override brake lights going to this bulb so when you step on the brakes, one side lights up solid, the other side blinks, if your turn signals are on while stepping on the brakes.
Hence, if you tap into the brake light wire in the truck for a third brake light, you will get a solid brake light when braking only and an occasional blinking light, depending on which side's wire you tapped into, when flipping on the turn indicator.