The brake lights run through the turn signal switch. You can still have turn signal when the brake circuit in the switch fails (IIRC, been a long time).

25 years or so ago, the I lost the brake lights on my '71 Duster. I narrowed the problem to the turn signal switch in the steering column. I can't remember how now, but I did rig a jumper at the turn signal switch, and temporarily fed the brake lights off the front turn signal part of the switch. The brake lights then functioned normally... Only problem was the front turn signals also functioned as brake lights every time I pressed the brake pedal. But I did then have good rear brake lights until I got the turn signal switch replaced.