Don't chase your tail trying to find the problem in the add-on central stop light. That's not causing your problem. The reason why the central stop light isn't coming on is because the logic circuit built into it detects power on only the one stop light (left
or right) that's working. That one-side-only condition is what the logic circuit uses to decide you're using turn signals rather than stepping on the brake (therefore not lighting up the central stop light). These logic circuits are notoriously failure-prone, which is why it's better to skip 'em and just run wire from a
good central stop light all the way to the stop light switch (like
this).
But your central stop light will probably start working again when you fix the actual problem, which is a faulty turn signal switch (or, less likely, a faulty wire from the turn signal switch to the right stop light). The power for the stop lights goes
through the turn signal switch; that's what allows the one filament on each side to serve both stop light and turn signal functions. Failure in the T/S switch takes out both functions on one side. The reason why the taillight still works on that side is it's a different filament (2 filaments, 1 bulb) fed by a different wire connected to a different circuit.
The '60-'61 turn signal switch is a lot more complicated and expensive than the '62-up switches, and retrofitment isn't easily possible without swapping the whole steering column. The '60-'61 turn signal switch shows up from time to time on eBay—
see for yourself.
