I’m not an electrical engineer so forgive me if this is a dumb question. Is it possible to wire it in such a way where the power actually travels through the LED, so if the LED ever stopped functioning or lost connection the webcam would literally receive no power?
Yes it's possible, as LEDs are indeed diodes, but done naively it would be a very bright LED, and it would tend to flicker as the current draw from the camera varies. It would be a hassle in general.
The LED could still always fail short, and then you're back to having a broken indicator again.
