This does not follow. You could make it an argument that essential services should be paid enough to live close to work, which would satisfy the criteria that you've laid out.
The most you can conclude from what you've laid out is that something is wrong with the current system. Which, I think very few people would disagree with.
Expensive housing is a problem for everyone even the people who own their home
I feel like I misunderstood something here. I presume the police/firefighters were needed to live near the community because they might need to respond in an emergency. Or possibly for community relations for the police (I know some jurisdictions require you to live in the community you police, at least in the states). If I'm wrong in this assumption, please correct me and supply some more context.
It's not automatically clear why the situation you've laid out includes the additional groups you've listed.
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.
And commuting has a cost too, driving an hour or two each way is expensive
Expensive housing is driving up the cost of everything because, everyone in literally every job is having their wealth extracted by landlords or into buying homes
So yes, home prices means taxes are going way up because all services are now far more expensive to provide because wages need to go up because the col for workers is going up because the businesses they rely on need to pay higher wages and here we are in a vicious cycle
It really doesn’t matter how it’s solved in the long run, but it needed a to be solved
Which seems like a problem that would solve itself eventually. If people's quality of life gets too low, I'd expect a decent chunk of them to move, opening up housing for people who can't/won't move.
There's some downsides to that strategy, so the city may not want to go that route, but as someone in a low CoL area, I don't really have a say in the matter (nor should I, so long as people aren't asking me to subsidize high CoL areas).