I forget the title but there was a Superman game on original Xbox that gave Metropolis a health bar, Not Superman.
Beating up bad guys, stopping disasters and completing missions helped the city survive. Getting hit meant a few seconds where you were inactive and the city might take more damage.
It was a pretty inspired idea honestly. I don’t think the game was excellent but good design choice.
I’m too old for it to be PS2, but “Mick and Mack: Global Gladiators”. Spent at lot of time on that for a game that is an ad for McDonalds.