I feel this idea of if you are attacked you have a right to take a couple of cities is a gamer concept.
I agree, the concept of righteous reparation is in our minds. It wasn't put in the game, in the AI. Now they incorporated part of this in the grievance system.
If you have units near/in their territory while not at war, they gather evidence and get angry. When you do so in war, nothing happens. You can shuffle around as much as you want, pillage and crush their units and break walls and there will be no complaints. However the moment you take a city, it is registered and remembered forever... However if the city rebels through loyalty and is taken by force, none of this happens.
The conquest emergency is called upon conquering a city. It seems it does not depend on anything else. In the stream, Carl certainly had enough grievances banked against Hungary and wasn't in the top score. Also, the emergency was called upon a smaller city, Miskolc conquered first, before Obuda. I believe the process is set up differently.
In my recent games I never had any emergency. Granted, I play king and am usually quite peaceful, but there was no emergency against AIs! The most common emergency I got is religious - when I convert a holy city, before a city state capture (with one enemy being the convertee). The AI pretty much never does any of that.
I wonder why are the emergencies called when I never get them, as I am usually leading the boards.(!)
Why is the AI hesitant to initiate an emergency. (!)
Why was the emergency in live stream supported by Inca and Canada but not Spain.(!)
The conclusion is: Emergencies are declared when a minor power is attempting to take on a greater power (score-wise).
Take that in.
Emergencies are not events to prevent snowballs, nor balance. They are tools to hamper the catch-up, or to challenge them. You want to get into the elite club? Show us in a war and perhaps you get rewarded!
Since it is based on score, the lesser civs do not vote or join against you. They are programmed to not attempt to take on greater strength enemies, as they fear retaliation from the powerful. However if the cards are reversed, they will not hesitate to declare surprise wars, join emergencies and send troops (as Canada did in the stream).
It makes sense historically. Islam conquering Christian cities? Crusade! Ottomans attacking Hungary? Let's ally (except for France) against them! Soviets in Europe? Cold war!
Sometimes it was different (compare Napoleon)
I am shortened of these experiences as I usually play efficiently and lead the scoreboards. This is why I can easily gobble small civs and be denounced and never declared war on forever. The strategy is to never challenge the big guys openly. Get bigger and you can do whatever you want. The AI will not stop you.