We don't have a SH obsession, the UK builds hardly any of it, Housing Association properties isn't the same at all, its still in the hands of private companies, many of whom offer shockingly bad service and still charge high rents.
Instead we've effectively privatised Social housing, with tax payers money going to LLs at far greater rates then if we still used SH.
If we tried to cap rents for LLs, they'd sell up, leaving people without the means to buy with no where to live, putting more strain on councils to house them.
In fact a council near me has borrowed a great deal of money to buy up private housing to reduce the costs to the council of supporting families in temp housing etc.
Houses have a minimum cost, the more we build, the greater demand on land, materials, planning and workers, which then increases that min cost.
SH means lower rents, lower rent support but perhaps just as importantly, it means people have more money in their pockets, so less need for in work benefits and more money to spend in the local economy too.
People can still move around the country as they did previously, with council house swaps.