* The NPC calls for his Captain to arrest this fool. The Captain (who suddenly becomes the intersection of the PCs successful check, the Baron's Flaw, and the setting at large) pauses and says "...he says nothing different than what the people are saying in the safety and privacy of their homes sir...arresting him will not sanitize that image...it will further sully it." After which, the Baron clearly shrinks and blanches while he gathers himself (after which the GM will either reframe the conflict with his next move from the Baron or another NPC present, or a player can make a follow-up move to reframe things).
The fact that you've invoked "aimed at the Captain" is a not-so-subtle indicator that you're not quite understanding how the dynamics of this work. Neither the player in their action declaration nor the PC within the fiction are "aiming their words at the Captain." What is happening is deft GMing. You need an emergent consequence which honors the players success while simultaneously honoring the nature of the situation and the component parts of the fiction. The gamestate and the fiction need to change positively. How are those two changed positively? The GM evolves the post-resolution fiction to put the Baron and the Captain at odds, thus stealing some of the Baron's boldness and, in the case of 5e D&D, revealing the nature of their relationship, thus providing an asset (the Bond) for the players to deploy in subsequent action declarations.
PC declares (roughly) "I try to talk the Mad Tyrant out of arresting us."
GM says, "The Captain says ..."
So, having the PC's success on a check affect a character other than the one the player intended--and declared--is ... an awful lot like taking control of the character away from the player.
Also, from the description of the event, the Captain wasn't there to hear the insult, so all he knows is the Mad Tyrant is ordering the PCs taken away in irons.
Sorry, no sale.
Huh?
In the OP: the player declares an action aimed at the tyrant (the insult), the action fails to achieve some sort of success (eg cowing the tyrant - we don't know whether the GM decided that by fiat or called for a check that failed), and then the guards turn up. The guards are part of the narration of failure.
In Manbearcat's example: the player declares an action aimed at the tyran (the insult), the action
achives success (ie changes the mind of the tyrant, by cowing him) and the GM uses the guards as part of the narration of that change of mind - the success manifests via the trusted lieutenant whose words induce shame. As he puts it, "You need an emergent consequence which honors the players success while simultaneously honoring the nature of the situation and the component parts of the fiction."
This is no different, its basic structure, from narrating a successful attack as invovling the enemy stumbling on a stone and loosing his/her footing. Or as per my Prince Valiant post not far upthread, narrating the lance splintering and a shard of wood going through the visor split and into the brain of the NPC knight, killing him.
Manbearcat calls it
deft GMing, and is correct. The GM takes control of that stuff which is his/hers - the NPCs - and uses those to change the ficiton in a way that honours the player's success. It's pretty much the opposite of what I see in the OP and much of the ensuing discussion, which involves the GM focusing intently on only one aspect of his/her stuff - the tyrant - and using that to ensure that the player can't get what s/he wants out of the situation.
My reference much earlier to the possibility of the guards being drunk, or disloyal in the fact of yet another manifestation of madness, are in the same ballpark as what Manbearcat has suggested. I think it's not a coincidence that we've both fastened on the guards: because these are crucial for the tyrant to actually act in the scene, but within the context of the fiction they are human beings with their own motivations which need not conform at all to those of their ostensible master. (Qv Emoer and Faramir disobeying direct instructions from their lords and kinsmen to detain strangers.)
EDIT: some of this was already said by
@Ovinomancer upthread.
Also saw this:
Conversely, overuse can lead to scenarios where there is no fail state or real consequences. Work with the Baron and you get what you need. Anger him and you still get what you need.
You seem to be ignoring the difference between successful and failed checks.
@Manbearcat was describing a possible narration of a successful check.