In my professional experience prior to the current Labour government, in my LA - housing worked like this…
Refugees assigned to an area under an agreed LA ‘quota’.
Refugees assigned to an area within the LA. Towns usually, rather than the isolation of rural houses.
LA made a conscious and considered decision to not assig LA housing. Very conscious of the antagonism in some local communities that this would cause.
Private rentals sought.
Home Office funding limits/allowance/cap per month did not match local private rents.
Local landlords appealed to, to rent within the Home Office cap.
Some landlords keen for altruistic purposes, some for security of the rental (regular payment and longer term contract - for instance Syrian refugees were allowed five years here, through the resettlement programme and then expected to return to Syria to support developments in their own country).
Some housing poor quality if landlords couldn't be found from the above. ( one was a huge Victorian house which was rented but had to house three families)
If no rental could be found, the families had to be considered elsewhere within the LA.
The difficulty with the Home Office cap on rents is that the cheaper areas of an LA receive a higher proportion of refugees than other areas putting some pressure on services.