In liverepeater origin/edge scenario only origin knows if stream is available (is being published). Edge tries to reconnect continuously and player doesn't get that error message (doesn't receive any event). At least Flash (RTMP) and HLS behave like that.
That agrees with:
I tried both, on-demand and persistent, modes.

Is there a way to notify player that the live stream isn't available when player connects to the edge?

I'm thinking about creating a HTTPProvider endpoint on the origin, and let player ask it, but an edge can be connected to one ore more origins that the player might not have the knowledge of (and it shouldn't have).
Maybe disabling edge's reconnection? Are there any disadvantages in that approach?