Sorry but that is not how that works. That is the brake lights are not "fed" through that switch. Either you had some other loose connection that you accidently "fixed" with all the wiggling, or that switch was shorting, but I don't know how that could be, as you should have smelled something
Here is how the brake lights work: Brake light switch power is ALWAYS hot from the fuse block, and when you tromp on the pedal, that switch of course activates. IF the turn signal switch is centered, the TS switch routes the power FROM the brake switch TO both of the rear lights. If the TS switch is making either turn, the TS switch disconnects the proper brake lamp and routes it to the flasher. It keeps on routing the brake switch power to the correct lamp
HOW DOES the hazard flashers work?
The hazard flasher always gets hot power. That power goes to one terminal of the switch you showed. That leaves three terminals. One other goes to the left turn lamp wire, one other goes to the right turn lamp wire, and this leaves the last switch terminal. THAT terminal goes to the TS switch on the same terminal as the brake light switched power.
How does this work? With the hazard switch OFF, all 4 terminals are open, nothing is made to anything.
With the hazard switch on, ALL FOUR terminals are jumpered together. This causes hazard flasher power to route to both front lamps, and to the brake light switch line. With the TS switch centered, the TS switch then routes that hazard flasher power to the two rear brake lights.
