A definition of demand: Demand is an economic concept that relates to a consumer's desire to purchase goods and services and willingness to pay a specific price for them.
The poor do not have the willingness to pay a whole category of prices, as they are unable to finance it. Creating luxury accommodation can increase supply, but as the decrease in price does not necessarily put the older accommodation within reach, the owners of captial can choose to retain it.
If there are 10 houses available and there are 12 families looking to buy then the they will bid up and the 10 wealthiest families win. If you build 2 luxury homes, it doesn’t matter if poor families 11 and 12 can’t afford them, because 1 and 2 can, which makes them not bid on normal house 1, 2, …, n lowering all housing prices.
Poor people don't buy accommodations they rent them in your example where demands exceeds supply owners of accommodations can jack up prices to whatever the poor people can pay without living on a street corner hoovering up any benefit obtained in your example.
What will the wealthy family do with their 3 houses? If they rent them out that’s still good for the poor family (indeed perhaps they couldn’t afford to buy anyway).
This is circular reasoning because homes are rising in value due to lack of supply. With enough supply it would not make sense to sit on an empty home as it would not make a good investment.
Given rental vacancy rates of ~5.8% in the USA in 2022 it is more akin to a hypothetical of building a 12th and 13th house, while an 11th already lies unoccupied.
I am not sure in this more realistic case that building improves matters, unless an unhoused family can gain access to one.
1) You can't take national vacancy rates, RE is extremely local, as is homelessness.
2) Is it easier for unhoused to gain housing when vacancy is 1%/5%/10%/20% ?
For example, Manhattan has 2% vacancy. Even in peak covid dread it never got much above 4%. And you know what - pricing updated very quickly to get those units filled. Lots of tales of people getting crazy deals, including people I know. Of course when the world was no longer ending & everyone moved back.. prices snapped back.
So even a 2% to 4% move can have huge pricing impacts.
Imagine building enough in HCOL areas to move the needle.
The poor do not have the willingness to pay a whole category of prices, as they are unable to finance it. Creating luxury accommodation can increase supply, but as the decrease in price does not necessarily put the older accommodation within reach, the owners of captial can choose to retain it.