When it comes to AI I'm already at war with, I agree that these are kind of pointless, but if not I do appreciate them as chances for "diplomatic exchanges" with AI. In particular, blowing them off is a good chance to either piss them off (useful when there are defensive pacts floating around so you can either bypass theirs by provoking them to declare war on you, or bait them into triggering your own defensive pact) or check how willing they are to actually go to war with you (indicated by whether they swallow their words when you respond with "Get over it/You will pay for this/etc.").Serious version:
Needing to stare at the game and be ready to Esc out of these completely empty threats from literally every other AI, sometimes multiple times per AI(I think, I haven't actually tracked this) in the part of the game when the turns take longer anyway gets old fast, but I have to do it if I want the turns to continue. It'd be nice if this happened once per AI at most, and preferably only from AIs who are actually competing with me in a relevant way, not just those who want to be.
(To be honest, I personally wouldn't be upset if this interaction disappeared completely. I don't feel that it adds anything, especially for someone who has multiple thousands of hours clocked with the mod.)
I THINK player response to AI threats might also affect the attitudes of OTHER AI, but this part I'm not 100% sure on. At the very least, from time to time I notice "they are disappointed with your recent behaviour" maluses on friendly AI they I couldn't recall any possible reason for except perhaps me giving non-hostile responses to threats from AIs we both dislike.