… Two things can happen: Either the planet gets very, very close to the binary, suffering tidal disruption or being engulfed by one of the stars, or its orbit gets significantly perturbed by the binary to be eventually ejected from the system …

This could be described intuitively in my opinion as a repeated slingshot effect … the same way that would make so that tides on Earth push the Moon gradually further away while slowing down Earths’ rotation …

In these cases, would each star orbiter each other and the planet have an elliptical orbit around both them or would be other type of arrangements like the planet doing an eight shaped orbit between both stars?

This seems like an argument from ignorance as detection of exoplanets is very difficult compared to detection of stars (whether binary, the majority, or not.)

The way we do detect exoplanets would probably be hindered by binary star systems. Call it glare from the second sun? Masking the dips in brightness.

But hey maybe NGT will show up and give me a lecture.