I'll illustrate with 4 families.
Remember we are talking about a house that at market rate would cost £800/month to rent, but currently the council rents out for £500/month.
Family 1
They have no income, the council has to completely provide for their accommodation costs.
Now: council puts them in a house and charges nothing: cost to council £800 (£500 housing benefit, £300 in lost earnings)
My scheme: they pay nothing, council gives them £800 housing benefit (enough to choose to rent a council house or a privately owned house)
Difference: nothing
Family 2
They have a small income, and have been lucky enough to get a council house
Now: they pay £200 rent, so the cost to the council is £600 (£300 housing benefit, £300 in lost earnings)
My scheme: they pay £200 rent, cost to the council is £600 (£600 housing benefit, they can choose to rent a council house or a privately-owned house)
Difference: nothing
Family 3
They have a small income, and have not been lucky enough to get a council house
Now: I have no idea what happens here - do they only get the same amount of housing benefit as Family 2? If so does that mean they only have £500 (their £200 plus £300 housing benefit) to try to find a house with? This seems just BAD
My scheme: they pay £200 rent, council gives them £600 housing benefit, they can afford to rent a council house or a privately-owned house just like Family 2
Difference: possibly costs £300 more under my scheme, depending on what actually happens now
Family 4
They got a council house a while ago and have a decent income so don't receive any housing benefit, but they have stayed in the house because it is cheap
Now: no housing benefit, they pay £500, cost to council is £300 in lost earnings. They is a strong disincentive to move because all the privately-owned houses cost a lot more. This family's accommodation costs are less than Family 3 even though they earn more
My scheme: no housing benefit - they pay £800 (this will have gone up gradually as their earnings increased so there was no shock) - cost to council - nothing. They can afford to rent from the council or privately
Difference: my scheme is £300 cheaper, which should pay for the possible extra housing benefit that Family 3 need
The logic:
1 - Anyone receiving housing benefit who does not have a council house should still be able (based on their assessed income) to rent a house that is as good as the council house that they would be awarded if there was one available.
2 - The subsidised-ness of your house (housing benefit, cheaper rent, whatever) should gradually decline so that once you are earning a decent wage there is no dramatic leap to be made, you have just gradually moved into paying the market rate