Except Batman's saved Gotham and the world a few times. Presumably not being a billionaire would make that much harder. Personally, I think the continued existence of everyone we know and love is worth more than all of Bruce Wayne's wealth.
When someone plants a bomb that will destroy Gotham and Batman defuses and removed the bomb, what has Batman done to the city if you don't think he saved it?
Like, if I save a child from getting hit by a car today is the kid now immortal?
Not quite sure which incident you're talking about but Batman's saved Gotham and the world more than a few times. He mortally wounded Darkseid in the Final Crisis event, stopped a worldwide Joker plague in Endgame (and Gotham city from a similar fate in the 1989 movie) and so on. As a member of the Justice League, I imagine he's saved the universe a good few times too.