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No, Keith, housing benefits are paid to the claimant, who pays the landlord.
The Council would probably not pay housing benefits to many individual people living crowded in one small place, so it is more likely as I stated above, that many immigrants who live crowded are doing so to save on rent, whether they are on benefits or working.
My guess is that they claim one thing when asking for benefits or applying for a job (that they live in a house, or flat, or bed-sit), but then pay the landlord less rent in return for sharing the same room with other tenants.
I doubt they say to the Council "I share the room with two other people".
Housing benefits vary if one is living single, or married; living with a close relative might exclude benefits.
As for London, where rent is very high, living in garden sheds is probably an individual decision of people who have an income of sorts, be it benefits or wages.
I doubt that the people concerned officially state they live in a garden shed. They probably use the address of the landlord to receive letters, so the address could be: 1 Victoria Park, and not: the garden shed of 1 Victoria Park.
But in return for living in a garden shed, they do pay a lot less rent, and have more money to spend otherwise. If they share the shed with other tenants, they pay even less rent.
Now, if a Council were to give permission to build, say, 1,000 affordable houses for renting out, and then asked the people living crowded together at one address, or those residing in garden sheds, to move into these affordable houses, these people would likely turn the offer down, as they would be paying much more rent if they accepted.
So I think we should not lose the plot on this one.
What would likely happen, is that a thousand people/families would move into the 1,000 affordable houses, paying the rent, and then sub-let rooms to more immigrants, who in turn would be paying less rent because they are sharing with other tenants. We'd be back to base one, but with a much increased problem.