It's not just a lack of housing though - it's thirty years of failed housing policy starting out with right to buy, continuing with scrapping rent controls and removing security of tenure effectively meaning that landlords are pretty much unregulated (as they can just evict a tenant that asserts their paper rights), and continuing again with a slow-down in both social and private house building, all of which was done with the express intention of encouraging 'investment' (ie speculation) in what has become known as 'the housing market'.
It really does go that far back - in policy discussions during the tail end of the 1980s, that is what the Tories wanted to happen - for the private landlord sector to take over from the state in terms of the provision of rented housing, and to encourage more landlords into the market through the removal of tenants' rights and making them easy to evict. It wasn't an overnight event - the housing crash of the early 1990s was what really spurred it on, which was obviously something they didn't foresee - but that was the intention.
Now that we've got here, we can see it's a complete disaster - 1 in 3 private sector rental properties are of a substandard condition, almost 1 in 3 private sector tenants claim housing benefit; ie we have poorer quality rental stock that collectively costs the country an arm and a leg in what is effectively a state subsidy far outstripping the amount of money it would have cost to have today's tenants in decent, affordable state-owned accommodation.
I don't really know what the easy answer is, because as you say it would cost a lot to reverse the entire mess, and in the meantime we're spunking ??12bn a year on effectively nothing and the private sector landlords are collectively the main part of the problem, benefit as they do from light-touch regulation and the unquestioning hand of the state. But the hard answer, and the one that would reap long term benefits, is as you say to build more state housing. It would cost more initially, but the rent receipts in what is pretty much perpetuity would more than cover the initial cost many times over during the life of the properties.