> Because expensive housing means they all need to charge more meaning your fighter fighters and we’ll every Goverment employee needs a higher wage to account for it, meaning landlords can charge more meaning wages need to go up.. you see the problem here?

Not really. Other than that you seem to be implying that the categories of people covered need to be able to live in the city, without stating that, and we've only only established this as a requirement for groups that need to respond in an emergency. Necessity only seems to dictate that said groups are present for the parts of their jobs that require them to be present. Commuting seems like a viable option, which should have a depressive effect on housing prices.

Fundamentally, cities with strict zoning laws tend to require commuters (most cities in the west post ~WWII or so). Some petty-tyrant always ends up on the city council and building grinds to a halt. Unless you also want to severely curtail zoning and building regulations, I don't think the proposed public housing is even actually a long-term solution. And if you do want to curtail those, I'd expect it to not be needed very much.