I'm not 100% on this theory of mine, but you may've found the problem. So maybe not necessarily the problem, but certainly a significant one nonetheless. That brake light switch on the brake pedal needs replacement asap.
Now, I think it would stand to reason that rear turn signals utilize the high-output signal (brake light output) from the electronic flasher cube at the bottom of the dash next to the fuse panel. I would at minimum get a replacement brake light switch and start from there. I wouldn't worry so much about the column-mounted MFS for now.