Hmm i'm a bit puzzled about it.
I would definetly play a EU4/Vic2/HOI3 style of game with a world map and where you pick a nation, just like you described set in modern times and where the main focus is how to battle catastrophes like famine, drought, climate issies, refugees etc. Many paradox games has themes in them (CK2 is about crusades, Vic2 is about industrialization etc ) and i think that climate change is definetly a great theme for a game set in modern times. However, what i do not like the doomsday idea. I think it ruins the game idea a bit when you mix "apocalypse and end of the world" into the game.
Just to make it short, here's what i think could make your game better:
The first 200-300 years could be about "The battle against climate change". The goal could be trying to help and aid third world countries in africa and asia stop the expansion of deserts, trying to invest in green energy in your own country, and try to influence other nations to switch to green energy. After the first 200-300 years, depending how much money and recources you have used throughout your game to battle climate change, there will be different outcomes. As and example: If you have completely ignored climate change and have just focused on reserching new technology and improving living standards in your own country, then you will be more unpopular among other world leaders for not taking action, asuming you're a western country, you will also have a BIG refugee problem because the desert have expanded so much throughout the game, and natural disater events will be more frequent and more serious and catastrophic than before and high pollution would cause unhappiness among your population and will make you unpopular among your own people too.
Then the rest of the 200-100 years could be about "the conquest of space". The goal would be reserching technology, and trying to set up a space colonization program and trying to develop it so that around the end of the game, you would be able to set up colonies on the moon and mars.
This is of course my idea. But anyways, hope you appreciate the feedback and good luck with the game :)