I've done a couple of these, they were referred to 'staying put' arrangements. I don't think it's correct to see this as an either/or situation, just HB paying for the accommodation costs, the accommodation being arranged by the local authority as part of the support they are still providing for the individual.
https://www.gov.uk/government/publications/staying-put-arrangements-for-care-leavers-aged-18-years-and-above
https://www.thefosteringnetwork.org.uk/policy-practice/practice-information/staying-put
