What makes the phrase “to be continued.” so idiomatic?

The passive ones (be continued) are fine. The middle (continue) uses are less common, but you do find them (eg “The story continues tomorrow”).

But none of them are very common. What makes to be continued an idiom is that it is used as one. Notice that, unlike your other examples, it doesn’t need a subject (or verb): it is common to see To be continued at the end of an episode.

Your question has an implicit “why?” Almost all “why?” questions about language have only one answer: “Because that is how it is”. You can often explain how something came to be the way it is, but almost never why it took that path rather than any other.