In somebody else's backyard. That's why you have concentrated areas of poverty. The government used to enforce this through "red lining" but since that has now fallen out of favor the language used to oppose these projects is now based around environmental impacts (the schools are full, the traffic is bad, it's a potentially toxic site, etc).

You forgot 'it'll ruin the character of the neighbourhood.'

What if those things are all true?

It's probably true, however, what is the correct point to "freeze" all development?

Most of silicon valley before the tech boom was orchards. Before the orchards, it was undeveloped land.

An argument could be made that development should have been frozen when it was orchards since it provided food, farming jobs, and bucolic scenery.

Environmentalists perhaps would prefer that even the orchards shouldn't have built and that the land was best kept as natural habitat.

You leave tens of thousands of people to run around trashing public spaces and consuming ~100k each year in emergency healthcare and maintenance costs to repair the damage they do, and that number keeps going up as the property bubble keeps going up.

Perhaps auction it off tokens to not have it in your backyard. All the proceeds of the auction goes to that place. Keep auctioning off the tokens until some place decides it'll be worth the money.

The result will be that the wealthy neighborhoods will never host the shelters, just like now.

But unlike now, they'll be paying the other neighborhood for that privilege instead of getting it at a discount via being able to lobby.

Sounds like an improvement to me.

As you point out they don't now. The proposal at least has the merit of extracting extra taxes from the NIMBYs.

They don't care where it is, so long as it is not in their back yard. As you have correctly pointed out, this is mostly what everybody says, in which case the obvious points: the project does not get created.