Have the city get worse and worse is a cool idea, though it would be better to base it on 'events' (i.e. doing a quest or similar type of progress quietly increases the distress stage) rather than with a running timer as you mentioned. A running timer punishes players who like to take it slow and enjoy the game and, vice versa, will cause cause players who are just rushing the main story to miss out on most of this interesting dynamic (because they'll be done so quickly). Also, imagine leaving the game idling while you go out for a couple hours.. lmao. Most of this also applies to step counters or any other kind of 'efficiency tracking' method. Making it event based also lets you make it so that (if you want) the player can't possibly complete all the quests available unless they are willing to stay in hell city lol.
Also, I request public bondage events!
True, I was never keen on time limits, but it's good to mix things up. Maybe as the main quests progresses, the city could get worse, as you suggested. Or I could find some happy medium: a really long slow burning timer, but certain quests can exacerbate or even temporarily halt the progression.
Public bondage events? Like captured would be heroes? People in gibbets? Damn right! I like the way you think~