This is a little pedantic, but "begs the question" doesn't really mean "raises the question" even literally. It sounds weird to say that someone "begs a rebuke"

This is a little pedantic, but "begs the question" doesn't really mean "raises the question" even literally. It sounds weird to say that someone "begs a rebuke" when they really deserve a rebuke. Even more literally, you don't say that someone on the street "begs a dollar" -- they beg for a dollar. Just as you said, some situations beg for those questions to be asked (or perhaps they beg that they be asked); they don't simply beg them.