What about a mechanic that rewards you for managing your housing and food, but doesn't require it.
I'm not sure why "houses" is a tech you have to rediscover (people in this world can build Inns out in the wastes, housing can't be a huge stretch), so if those first three levels (slums, huts & houses) were functions of city prosperity (education, wealth and prestige) that your citizens built automatically then basic housing could be ignored. You'd then have three options a) ignore housing entirely, allowing your populations to be a scrabbling rabble
build prestige and educational buildings to let the people build houses, being happier and more productive but taking away from the tiles you could otherwise build (since you built educational, etc.) c) take even a few more tiles and build high-quality houses. In fact, you could capitalize on high-quality housing by making it a requirement to attract talented adventurers or such (or making for a discount in their price), etc.
A similar mechanic for food would work (if it were per city or you didn't have caravans running), lowering the output capacity of your city, lowering prestige and (if it were bad enough), actually causing starvation die-off.
As for setting related things, e.g. nobles, inns, etc... Well, nobility isn't money it's power and those who had power before the cataclysm and whose family maintained their power would likely still hold on to titles and local spheres of influence. In fact, I would welcome some of them challenging your authority (and I seem to recall "neutral" city-states being a possibility).
And of course inns would exist - in fact, they might well be a nexus for founding small villages and such - as would any number of common structures. Along those lines, rather than random Inns scattered about, if they were villages and/or estates (that would eventually fall under your sway if/as your territory expands and preferably named), I think it'd make far more sense. I wouldn't be surprised if calling them Inns was an intentional beta thing (rather than giving away info from the campaign).