TIL that a high school in Pennsylvania issued an Apple MacBook to each of its 2,306 students, then remotely activated the webcams to spy on the students at home.
An LED is pretty damn simple and there's no reason it would even be possible to control the voltage going to it. May as well worry about hackers finding a way to make a physical shutter transparent.
My current work laptop has a shutter built in which heavily blurs the camera, so it's relatively obvious why your camera isn't working but you still get some level of privacy
that could work, or perhaps have a cut-out on the camera cover that blurs all light going in and has the words written on it "shutter closed" or something. Digital way built into the driver is probably easiest.