In Garner's Modern American Usage (Oxford 2003), Garner says, "As a relative pronoun or relative adverb, that can be suppressed in any number of constructions... {b}ut in formal writing that is often ill-advisedly omitted, creating a [miscue]," while also acknowledging that those who delete it excessively seem to be overreacting to others who use it excessively. He then quotes a sentence with five thats which he then revises so that it uses one.
An excellent example where "that" is very useful is "the Karcher group claimed the property was worth $2 million" when "claimed the property" is a common phrase that makes it easy to momentarily misread the meaning. While less extreme, "Alice knew the police were coming" makes it possible to wonder if "that" or "who" was omitted. If Alice knew Dick and Jane the cops, then the intent might have been to say "Alice knew the police who were coming." To put it another way, "Alice knew the police" could be a complete clause that allows the reader to sense completion and look for the next unit of meaning. "Alice knew that" can't and signals that the reader needs to read on for the sense. As you say, it can be a necessary connector.
It seems like there ought to be a simple rule based on something definite, but I don't know of one and Garner doesn't seem to either so, as others have said, it seems to be a matter of taste (which may vary between English and American audiences) and probably shouldn't be judged in isolation. If the context makes the meaning clear, if it's unlikely to produce a "claimed the property" sort of misread, and if it improves the flow of the writing, then omit it. Otherwise, include it. While I might omit it in context, in the above isolated case I'd leave it in. Alice knew the fact, not the people, so she knew 'that' the police were coming.
