While 'no expectation of privacy in public spaces' is an American thing, people in general accept any number otherwise awful things as long as they are part of the status quo. I mean, I can go on, but one only has to look at the history of man to look for appropriate examples.
i'm an american and it is is exactly stuff like this that causes me to avoid using services like this like the plague if i can avoid them.
despite living here, privacy to me is important. if i were this woman, i'd not let it go at all - don't give these scumbag companies agency to do what they want with impunity.
Suppose a driver complains to Lyft about a passenger, or a passenger complaints to Lyft about a driver. Then it would be useful for Lyft to have audio and video of the car ride to substantiate the claims.