Despite your claims about "lost money" because social housing, in your opinion, could cost more, it's not social housing tenants that cost the exchequer a lot in housing benefit bills. Social housing keeps housing benefit bills lower than they would be if everyone were in private rentals.
those are state assets, and letting them out to people for below market rates is in opportunity cost. If the state did that with Tesco, i.e. rent them land for a lower rate than someone else was willing to pay, there would be outrage (and rightfully so). So, just because social housing is "paid off" doesn't mean that people there aren't preferentially subsidised. The state could (besides legal issues), let those at a higher rate and build more houses with the money.
The problem is that you include in "need" as "need to live in zone 1". that's not a "need"; that's a "want". a want that about 5 million people have.
And no, not everyone who can't get social housing is forced to move to Milton Keynes, commute two hours to London for slightly above minimum wage (the "slightly above" part being to pay for travel costs) and pay for twelve hours' childcare per workday. I mean, they just aren't. That just does not happen.
I am not sure what bubble you live in, but millions of people (literally) commute when they would rather not. I am not talking just about the poor, I am talking about everyone who would prefer to live more centrally but can't because of cost. if you don't think this describes millions of people, than you haven't read the news lately.
It's not even part of the solution.
then we don't agree on what the problem is. you possibly think the problem is that poor people are gentrified out of their neighbourhoods. That's a small part of the problem. The problem, as I see it, is that most of us millions of Londoners are subject to massive rent and property costs, and most of us are not benefiting from it, even owners. It's completely out of touch with reality. Doctors can't afford to live in central London. HB and social housing isn't going to solve that problem, as demonstrated by years and years of having it. In fact, it makes the problem worse (HB pushes up rents).