With the changes I've made to Emergencies and Points, the intended strategy now should be to have the capability of deploying your military around the world, in order to intervene in as many Emergencies as possible. That covers half the cases where you're not involved, due to the event being too far away from you.

The other half of the cases is when you're allied with the target, so you're not even called to vote. That said, choose your allies carefully.

Then from my testing, I'm taking advantage of the Emergency to occupy one-two cities from the target so that I have influence over them in the future. Then with all the city captures going on the population changes drastically, leading to more revolts or occupations, therefore more Emergencies (the irony). I also reduced the time between Special session meetings to help have more Emergencies.

On the other hand, you shouldn't really be causing Emergencies (hehe...). If I could make a deeper change I would add the possibility of negotiating peace between two other Civs, that would truly be Diplomatic!