Any way to turn it off would defeat the entire purpose. It's annoying, but I'm still glad that it's there

You can turn it off for external displays, it requires a few steps because you're weakening privacy protection.

Check https://support.apple.com/en-gb/118449

I think that’s meant to be a security feature. If it can be toggled off in software, that lessens the trust that it reliably represents the state of recording.

Maybe a black circle sticker might help? I see people do that to block the camera. Should work just as well for the LED.

Or a projector..