I'm just trying to think through the implications of the theoretical scenario of disallowing the renting of housing. My goal would be to reduce wealth inequality - as rent is generally a wealth transfer from the poor to the rich - but this would pull on so many threads in the economy that I wanted some other opinions. So a builder or investor could own an apartment complex and sell the individual apartments, and the buyers could pay in installments that would be essentially the same as renting, from the perspective of the person living there, with the difference being that after 10 or 20 or 30 years the occupant would own the apartment and not owe anything but taxes and maintenance. If someone knows that they are only going to stay someplace for a year or two then the buying/selling process would need to be much simpler and cheaper to avoid all of the transaction costs and paperwork. How to distinguish apartments from hotels is another question.
If the required downpayment is substantial there will be fewer households if renting is illegal. This means the average number of people per household would be greater. Some people would be living in larger households as they save for a downpayment. If the required downpayment is zero then price volatility would imply that some home sellers have a debt from an earlier home ownership experience. If experience shows that debtors tend not to pay then the required downpayment will be substantial. The most likely outcome is substantial required downpayments and a greater average number of persons per household.
When there is an attempt to control the market in this respect, there must be a determination to make the control total.
The effect of outlawing landlord and tenant relations - which I think is what you have in mind by the "disallowing the renting of housing" - would be to force the majority out of relations of landlord and tenant, and into relations of mortgage-lender and borrower instead.
Rent would still be extracted, but it would be conceived of as "interest" - the charge levied for the rental of pure capital, which is used by the borrower to purchase the house - rather than as a charge levied for the rental of the house directly. It is also not necessary, with a mortgage, that the principal sum is ever repaid, so this path does not necessarily lead to ownership.
Already, through "buy-to-let" mortgages, the majority of rent is ultimately extracted not by the landlord, but by the mortgage-lender to which the landlord accounts in turn.
The main problem we currently have is that there is an absolute shortage of housing units relative to the number of households , allowing the asset-rich to benefit at the expense of the asset-poor, and any solution will involve forcing up the supply of units dramatically.
I too would be interested in gaining a fuller understanding of why such supply is short - shorter in many respects than might be expected in a functioning market, let alone relative to what state planning has been shown capable of delivering in the past.