Brake lights aren’t working.

"Applied power to the wire"

OK we are not there, we can't see, WHAT wire? The white, going to the TS switch? The power wire feeding TO the switch?

If you don't have constant power TO the switch that is PART OF YOUR ANSWER right there. Find out why. Fuse? Broken connection?

The remaining wire, usually white comes OUT of the "cold" or switch output, and leads to the TS switch. That terminal should have power anytime the switch / brake is pressed.

From there, the TS switch routes power to the rear lights. Connects them together with switch centered, and disconnects whatever lamp is signaling

THE HAZARD SWITCH MAY BE A CLUE. The lights go through that section of the switch. Try working/ wigging the hazard knob and see if the lights flash on or start to work.