I think I understand your problem. Civilization is kind of a game that encourages you to create objectives for yourself before setting up the game and than making an effort to reach those objectives. You actually can pick random civilization for starters and see where flow of time gets you, but much more fun is to have a plan beforehand. For example you decide that Elizabeth has to rule the world, and she has to be warmonger and in police state. Then set up your game and "make it so" (read this in Picard's voice).
Also you might want to adjust your timer for timer to flow faster (you can do it while setting up your game).
Don't know about how much immersion you can get, but civs can get pretty intense for me, especially if things go not exactly my way all the time.
In that game though, I have an easier time connecting the mechanics and the "story" of my game. When I do things and when stuff happens I can easily imagine what's actually going on down there in the castles and on the fields. Civ V feels so much more abstracted and I have a hard time getting invested in it despite its beautiful surface. I love board games with a very strong theme that connects the visuals, game mechanics and my internal "story" and I guess it carries over to videogames as well.
