It is extremely unlikely (in the human terms - not possible) that LED would break in such a way that it transmits the power and not emit light at the same time.
Of course the LED can be installed in a parallel connection on the circuit, but I read the op statement as it is not the case.
"Let's devise a way to purposefully burn the LED in such a way that current can go through." We have to keep in mind that not everything has to be by accident :) I have no idea how likely it is to be possible, though, you've got a point.
Why did you read that it's not the case? I read the opposite. That if the LED broke, you'd notice it because your camera would be on and the light would be off.
I would bet on the LED being parallel - otherwise sending more current through it will burn more power, generate more heat, and likely cause it to wear faster.
I think the point though is that it's not software controlled in any way: powering on the camera lights up the LED, and there's no way to bypass that with only software. Or at least that's the claim.
Of course the LED can be installed in a parallel connection on the circuit, but I read the op statement as it is not the case.