If the third brake light works; this rules out the brake light switch; since that switch activates both the main brake lights and the third brake light. I believe it also rules out the steering column electronics. The most likely issue is disconnected or corroded ground wires for the brake light assemblies. There should be a dedicated ground wire from the brake light housing on each side to a bolt in the body or vehicle frame. These wires are vulnerable to being broken, disconnected, or developing corrosion at the connections. One way to test the ground circuit is to have someone sit in the vehicle and hold the brake pedal down; while you take a jumper wire which has one end connected to a known good ground point (you may need to run a long ground wire all the way back from the battery, if there is no proven good ground point in the rear of the vehicle) and touch the free end of that wire to the housing for the brake light on each side. If the brake light comes on when you touch the ground wire to the housing; this proves that the housing is not adequately grounded.
The regular brake lights draw more current than most other rear lights (including the 3rd brake light). This often causes ground faults to not interfere with the other lights; but to prevent the brake lights from working.
Some vehicles may be built without a dedicated ground wire from the brake light assemblies; and just use the moutning bolts for the light assemblies to carry the ground current through the body panels to the front of the vehicle and the battery ground cable. This can work while the vehicle is new; but it often fails when a vehicle is used in an environment where the roads are salted in winter; or if the vehicle is operated near the ocean. Salt can corrode the electrical connection between body panels, and in that way, break the ground circuit. It also can act up if the vehicle has been repainted, and paint overspray insulates the body from the mounting bolts for the light housings. In those circumstances; it sometimes becomes necessary to run a long dedicated ground wire through or under the vehicle from the battery to the rear light housings. 12 gauge stranded wire is recommended for this purpose; with appropriately sized ring terminals crimped to each end.