There is a third way. You can reduce demand in other ways, like steep taxes on second and third homes. This releases pressure on the rental market by enabling some renters to afford to buy rather than rent.
The properties people purchase as second and third homes aren't going to make a difference to the people struggling to afford 1 bedroom apartments.
Focusing on second and third homes is just a way to punish people wealthy enough to afford multiple homes. While punitive taxes against other classes are attractive to certain parts of the voter base, it doesn't actually improve the situation for people looking for cheap apartments at all.
It seems like if you add supply to one area of the housing market like the one above, there is a group of people ready to take advantage of that supply. 3bdroom apartment renters buy a home, then 2bdroom apartment renters get a bigger place that's affordable and meets their needs.
Assuming this works exactly like intended, that 1 family with two or three houses now only owns one and 2 other families get a house, then you're freeing up supply across the whole range of properties.
They will. As fewer homes are allocated to current homeowners, they will be instead available to renters. With fewer renters on the market, the cost of renting will decrease.