Two-tier types of tenancy cannot be fair: One person, with low income during the period s/he is applying for help, will be given a house for the rest of his/her life, all at a greatly subsidised rent, and despite the fact s/he might become rich, after moving in. Another person, with the same income, will get a private tenancy, with a rent many times higher, and no life-long tenancy.
A fair system would start at the beginning, asking "What must government do, to ensure all citizens have a roof?" Limit the number of citizens, to a reasonably sustainable, affordable, feedable and houseable level, would be the first part of the answer. Next, ask if anything stops any one of the citizens from going to live in a home suited to his/her own needs, to an extent requiring government intervention. (Governments don't provide clothing, and food, which are even more essential than shelter.) If the reason some citizens cannot find housing is to do with price, then imbalance, of ever-increasing demand and finite supply, is the first and main thing to correct.
The day Housing Benefit began, should have been the day council tenancies were made fair and equal. In other words, the financially viable reasonable local rents should have been charged, making council houses a source of income for other vital council services, not a subsidised drain. The terms of tenancy should also have been made equal, in other words, the occupants should be able to be asked to leave, if the landlord needs to sell or renovate, or if the activities of occupants are detrimental to the property and/or to the neighbours.
If the home-seeker has a temporary difficulty of an income too low for the rent, then s/he, like all the nation's tenants, are able to apply for Housing Benefit.
However, they are not able to keep getting public-purse money the moment they have enough money to pay full rent. (Becoming a union boss, or M.P., for example) They also must not keep applying for Housing Benefit for a child, when the child has left. They also must not keep applying for Housing Benefit as a single person, when a partner has moved in with them, and should reasonably be expected to pay a share of the rent.