> You cannot send power to the camera without also sending power to the LED, which will in turn cause the LED to light up. Unless the LED is broken, in which case you will know because it will never light up.

You make multiple assumptions here

1) You assume that the during the time that passes between the LED breaking and the user noticing, there was not a single attack or a single blunder that caused the camera to turn on and record/capture something that was unintended.

2) You assume that the LED breaks deterministically. The LED can break randomly. Maybe it lights up when nothing is being recorded resulting in a false positive. The user has no way of differentiating between a false positive and a true positive which can result in unintended captures.

3) Similarly the LED can break in a way where it sometimes doesn't light up when something is being recorded even though power is always sent to the LED when the camera is on resulting in a false negative. Again, the user has no idea of differentiating between a false negative and a true negative.