As someone completely unfamiliar with the media landscape in Israel, is Haaretz the one most likely to investigate/publish something that would be embarrassing to government, regardless of which one is in power? Any others?
Haaretz is generally considered left wing, but also the most “serious” of Israeli newspapers. There’s no comparable newspaper on the right (like the WSJ is in the US), and it’s probably a a little further left in Israeli politics than what the NYT is in American politics.
Hmm. It is a hard question. I think the closest comparison to it would be UK's Guardian. They have an axe to grind and an ideological stance you can gleam from almost every article, but it is hard to say that they would not publish things from their preferred government.
I base my answer on about 5 year occasional reading when checking regional news.