'65 Dart No brake lights. Brake light switch failure?

The brake lamp wire doesn't route thru the turn signal switch in 1965.

Does so; that's how the brake and turn signal functions are provided by the same filament. Brake light power runs from +12v to the brake light switch, to the turn signal switch, to the left and right rear lights. The turn signal switch is designed to supply power to the rear lights either via the brake light switch or via the turn signal flasher, depending on the position of the lever:

Left turn: left rear lamp fed from flasher; right rear lamp fed from brake light switch (if it's sending power because your foot's on the brake)

Right turn: right rear lamp fed from flasher; left rear lamp fed from brake light switch (if it's sending power because your foot's on the brake)

No turn: both rear lamps fed from brake light switch (if it's sending power because your foot's on the brake).

This is why, with the ignition off and the brake and turn signal switches in good working order, stepping on the brake and moving the turn signal lever to the left or right position will kill the left or right brake light: the turn signal switch is breaking that lamp's brake light power feed and connecting that lamp's turn signal flasher feed, but there's nothing coming from the flasher because that's an ignition-switched circuit.

You needn't believe me on this, but you'd get in a knock-down/drag-'em-out quarrel about it with the factory wiring diagrams.

It did by the time of my 1969 Dart. You can tell because the replaceable plastic cam in later cars has 2 wires on it vs just-plastic in earlier ones.

No, those two wires you see on the 1969 switch are part of the lane-change feature added around that time (flash as long as you hold the turn signal lever slightly up or down from the neutral position, without clicking it into the left or right position). And those two wires you're talking about aren't necessarily indicative; the final-revision version of the switch without the lane-change feature (this one) also has two wires, one red and one white, on the underside of the cam.

A faulty turn signal switch on an A-body of any year can certainly cause one or both brake lights not to light up, even if the brake light switch is good.

OP, if you wind up needing a new turn signal switch, shop carefully—there's a lot of poorly-made junk on the market; mine don't suck. :-D