Consider that I might want to record an encounter with law enforcement on the sly, and also want to keep my phone locked to make it harder to tamper with if it's seized.
Regardless, an app like Firefox should not be doing this, or even offer it as an option. There are certainly legitimate use cases even for Firefox, but I think with a web browser there's too much possibility for abuse and security/privacy issues to make it worth it.
Why shouldn't I be able to use a website called imgettingpulledover.com on Firefox instead of an app to do the same thing that you just described?
EDIT: posted this before the edit, edited version makes sense.
I disagree with the permission part, I hate how everyone's solution to any problem is to just add a toggle or another level of complexity to the app.
But the solution to the first part is easy and something that already exists all over Android. Simply have a persistent notification saying "The camera is currently active" which can't be dismissed until the camera is turned off. This is used in newer version of Android for any background task doing non-trivial work, really.
Unfortunately, I doubt Android will ever be able to rely on a separate visual indicator of recording, since that's another hardware component and probably hard (if not impossible) for Android to enforce.
If there's not, the app looses access to the camera while the app is backgrounded/the phone is locked.
One feature that’s been in the iPad for a few years and coming to the iPhone in the fall is picture in picture where a video call doesn’t take up the entire screen and as long as PIP is enabled, you can have your video call minimized and do something else.