What would happen is that developers would buy up blocks of housing, demolish them all, and build more luxury units.
The average owner of a single-family home can't afford to demolish it and build a new block of flats. All this would accomplish is to make it financially feasible to buy up single-family homes and build more luxury units in them, concentrating even more wealth in the hands of property developers.
This is basically exactly what's happening in Vancouver; build a condo tower that starts at $1m/unit, it gets bought by either the rich elite or property speculators (or individuals attempting to 'hide' their wealth in other countries), and then sits idle. Even with rent control, new housing being built is predominantly geared towards the upper class, or the top end of the upper middle class. Property values go up, and the only thing keeping rents even remotely reasonable is the rent controls.
Then, because rent is controlled, property values are kept lower than they would otherwise be, because the 'buy-and-rent' landlords can't pay for a mortgage that costs less than market rates for rent, so fewer landlord tycoons buy as many units. This means lower mortgages and more supply for people who actually want to purchase a home and have actual security.
Your proposal would still concentrate power in the hands of developers and landlords; developers would buy up all the houses that would sell, knock them down to build rental units, and then, if rents risked dropping 'too low', they'd stop building for a while and just hold on to the properties.