Why the fuck isn’t there just a simple status LED that is on the same circuit as the camera?
Because cameras aren't simple on-off devices powered by a single wire, that's why. It's always got power, and it's turned "on" (send image data over the data bus) and "off" (do not send data) by software commands over the same data bus.
So the most convenient solution is then have the camera IC have an output that can drive an indicator light. And as camera ICs are basically full computers in their own right, they can be reprogrammed so that they don't turn on that output.
End result is that you are much better off either having a physical cover over the camera lens, or having a USB camera that you can unplug.
Framework laptops solve this by having physical switches for the camera and microphone at the top of edge of the screen. Can’t get safer than that except for physical removal.
Anyway I think that a Webcam cover should be a default on webcams, not only for privacy, but also for th same reason why Cams and binoculars have caps by default, to protect the lens. If you use a tape, to avoid glue on the lens, you can cover the intern part which is over the lens with a piece of paper