If you look at your schematic I think you will find that the flasher is before the turn signal switch and that's how the proper ts light is determined. It's also why using any input from the flasher for on/off short of a trigger for a latching relay won't work.
Why not a switch? I'm almost always in the right lane, if I need to look at something the mirror won't show it would be to the left for a lane change. I would rather the camera be on all the time. Or leave both cameras on constantly with two monitors, they're not that expensive.