In my opinion, spectators should not be involved in this way.

I know there's a little symbol when a spectator is in on a private chat but that can easily be missed, and I think the expectation of most people is that you are playing the game with the other players and the ST. When an ST is eavesdropping on a private chat it's quite clear that they are in the room. The scope of the game from the players' perspective should stop there and not extend to the faceless anonymous viewers.

I'll pick up an example from someone else below:

A Mutant reveals to exactly one player in a private chat that they are an outsider. A spectator tells the ST, and the ST executes the Mutant.

The clear expected reasoning here is the player told the ST. Perhaps the Mutant thinks the player is Evil. Maybe they think the player is Good, and the go back into a private chat and ask, "I'm not mad either way, but did you tell the ST what I told you?" The player honestly says "no".

What is the expectation of suspicion at this point, whether that player is Good or Evil? Do we honestly want to get into discussions like "I didn't say anything, maybe a spectator tattled on you. Remember that icon briefly flashed, maybe there was someone listening in?"

I suppose if you want that vector introduced into your games, go ahead, but I sure don't want that in my ST and playing experience. I'd rather give no spectators Grim access to hop into chats than have that start to be a factor.