Lack of supply is only part of the problem, and not a very important part at that.
Between 1945-1980, home ownership doubled. Why? Because house prices were essentially regulated; kept low and stable by a combination of government policies:
Briefly, there was always a cheaper alternative (renting/council house), and people were less willing (or able) to borrow vast sums of money to buy houses if tax liabilities would follow.
Thus, house prices had to stay low to meet affordability, and rents were cheap, so young couples with one average income between them could easily save up for a small deposit on a house, and afford a 10-year mortgage on the rest.
Over the years since, all these policies were steadily removed or watered down, most egregiously by the Thatcher government. Rent controls scrapped, bank lending almost completely deregulated, council house building stopped (after all the nicest ones were sold off), Schedule A taxation and Domestic rates replaced by a Poll Tax (later rebranded as Council Tax).
The Blair government continued the trend, did nothing to reverse it, probably because rising house prices made the middle-class voters he craved feel richer.
The burden of taxes shifted off property and now falls most ponderously on incomes and productivity, with predictably bad results for wages and employment.
Property is now an unregulated tax haven, and the entry fee (house prices) has shot up accordingly.
If the Tories had any interest in deflating the housing bubble, they would be looking at any of the above policies, or even better, at a tax on land values, which would avoid most of the obvious downsides of (for example) rent controls.
But they're not. Their only solution to the housing crisis so far is Help-to-buy. More lending! Even higher house prices in the medium term!
And why should they? Banks like high house prices; they can lend people more money over much longer periods. And wealthy landowners like high land values; they can charge more rent for the same building costs. Clearly, those are the people whose interests they represent.