If you look outside the US, you can get housing for a price that leaves enough of the 4000 for other expenses.
The easiest option is to connect some barren land to a city center with public transport in a climate zone that doesn’t need much insulation. But that just as a proof of concept. More clever options can be realized.
Right, they are a monopoly until there is competition. That’s OK. You tax them higher until it becomes profitable for a competitor. That’s not ‘out of business’, just high enough.
But you can also accept the monopoly if the offer is transparent and good enough.
The colluding is a problem. It is the problem. It’s unavoidable. In every system there is corruption. This cannot be solved but has to be dealt with case by case.
Link your end with the start. The top earners don’t have the income to finance it.
My point is not that UBI should be a tax on the rich but that regular citizens can finance it right now if they want it.
You hope that UBI comes for free. It won’t. The majority has to pay it with higher taxes, voluntarily or not. So if they want it then don’t wait for politicians but implement it right now.
Have you seen how housing prices rose when interest rates were low? Markets work that way because consumers outcompete each other, at least in the housing market. You need a surplus of supply, like the corn market, to keep costs low.
Like @espi wrote, you need fierce competition in all markets.