A couple of thoughts, if I may.
First, the cost of a house is hugely influenced by the cost of the land it sits upon. You could think that your average £300,000 house on a new housing estate costs perhaps £150,000 or less to build, and the rest is the cost of the land. (Made up figures, but the principle holds.) Farm land, without planning permission for houses, is about £10,000 / acre. With current densities, you get about 10 houses (on average) in an acre of land. i.e. £1,000 per house if built on farm land.
It is entirely within local authorities control, to buy up farm land at £10,000 / acre, grant itself planning permission on that land. And sell it to builders at £1,000 / house, with a contractual agreement on the retail prices the builder can charge. At a stroke, this could halve the cost of new builds across the country.
My second point relates to young people's (in)abiltity to buy houses. When my mum and dad were young, you had to save with a building society for years, in order to earn the privilege to meet the building society manager, and beg for a mortgage. Only to be told "come back next year and we'll see". This was the norm. People got married and lived in the spare room upstairs at their mums and dads, or their aunties or whatever. When they finally did manage to buy a house, they had no car, no TV, no fridge, no furniture. I was born in 1961 and we didn't have carpet until I was about 10. When I bought my first house in 1988, we sat on deck chairs in the lounge for 2 years watching TV on a 19" portable borrowed off my father-in-law, since we could not afford a TV or any furniture.
Kids today expect (or want) everything on day one. Now I do not resent them doing that, but I do think this idea that it is terrible on young kids today, is overplayed. Times were hard in the past too. These things go in cycles. At the moment house buying is out of the reach of many young people. A few years of house price stagnation (or even another crash) and a few years of wages growth, and they will be more affordable again. I am not saying there is no problem, merely wishing to put this into context. Life, and making ends meet, is a struggle for most young people, but it always has been.