You are right about the brake light circuit going through the turn signal switch, it does.
When I installed a new turn signal switch, tightening its fasteners too tight tweaked the body of the switch to the point the contacts didn't touch. No brake lights. I backed off the fasteners just to the point where everything started working and it's been a couple years now, no problems since. Maybe that's what's wrong with yours.
Since your turn signals work then you already proved rest of the brake light wiring is good including grounds. Time to pull the steering wheel.