You can't control price without increasing the supply. That's the point. No one wants to increase the supply of affordable homes, that's the issue, and it would take a National Resolution. To do what it takes to build them and (you are quite right) the associated infrastructure.
But, lacking this resolution, it's no use people banging on about how much the state spends on benefits because effectively a large chunk of the population (mostly working despite the propaganda) has to be subsidised. Many jobs are (objectively) financially unsustainable, so, therefore, the government has to make up the difference to enable people to live.
While our industry is generally below ideal levels of productivity, this is a long-standing problem. I remember Mr Heath banging on about it while I was still at school. No government has sorted it since and I doubt any will in the foreseeable future. Again, huge investment would be needed, but it's more likely we shall remain a low-wage, low-investment economy. Real wages are, if anything, declining rather than increasing for most people.
That being the case, the only way to square the circle is to fundamentally reduce the cost of housing, which means vastly increasing the supply. Alternatively, just accept that many people will always be propped up by benefits paid for out of tax.
I am not sanguine about anything happening BTW as this country is not very good at grasping nettles. We are much more likely to woman and whine about certain people 'living off the state' while ignoring the truth that many wealthier people are 'living off the state' thanks to their companies and properties being indirectly subsidised.