Absolutely, Trickle. We were very fortunate with our current house, because my Uncle is a landlord, so although a relative, we could demonstrate that he has a portfolio of properties, so therefore qualified for HB despite him being a (declared) relative.
However, even though he is very flexible and we were allowed to do anything with the property, there comes a point when you have adapted as much as the property can be adapted and it still doesn't meet the needs of your child.
The market rent isn't what private landlords rent houses for. That is an overinflated rate. The council housing is a realistic rental value.
WRT the over-simplification, I disagree.
Currently - rent £650, LHA £450, we pay £200.
New house - rent £450, LHA £250 we pay £200.
The LHA is council money, and the rent is going to the council. Therefore, the LHA is reduced by £200 but also goes back to the council and we pay the council £200 rather than my uncle.
The council goes from paying out £450 to receiving £200 (the LHA becomes cost-neutral..they are paying themselves so no gain and no loss).
Yes, they have costs associated with the house, but they would have that with whoever was living there.