If the private LL all sold up, house prices would fall. Some of those currently renting would then be able to buy. Also Housing Associations or councils could buy from the private LL.
Imagine 101 houses and 101 people. 1 person is homeless, 1 house is long-term empty, and 100 houses are occupied. 20 of the houses are privately rented to 20 tenants so 20 people own two homes. These houses are sold, and the price drops to a point where 20 people can buy. The houses and the people balance out. The houses are not left empty.
Some of the 20 LL have high mortgages, but they are now legally barred from letting the houses, so have to sell at a price below their mortgage, and go bankrupt. The banks take a hit on unpaid mortgage debt.
(This simplistic example assumes no new house building and no population growth.)
This will never happen because the government protects the banks.