The assumption in the OP (and supported by some posters n this thread) is that the main (possibly sole) reason that the system is "broken" is that there are too few houses and, further, that this shortage is due to the delinquency or malice of the Tories. This may be true in part. It is also true (as acknowledged in the BBC link - 250,000 new homes "needed" each year - and in a couple of posts here) that housing crisis is just as much due to an increase in demand.
A few illustrative examples taken from a recent report (on social integration, not housing per se):
"1.23. In the year ending December 2015, the ‘net’ immigration figure was 333,000 – but emigration does not really ‘cancel out’ immigration; it is the total churn in population that can alter the characteristics of a neighbourhood and the net figure of 333,000 reflected almost a million people in total arriving in or leaving the country over 12 months. Additionally, the placement of asylum seekers across the country – often in poorer communities – and the presence of an unknown number of illegal immigrants, adds to the level of change being experienced.
1.24. Higher birth rates among foreign born parents are also contributing to the growing diversity of the UK - while foreign born residents made up 13% of the population in 2011, 27% of births in 2014 were to mothers born outside the UK (predominantly to Polish, Pakistani and Indian mothers).
1.25. The impact of these changes is far reaching.
1.26. We were told on a visit to Sheffield that more than 6,000 people of Roma or Eastern European heritage (of which more than half are under the age of 17) live predominantly in one ward. The impact on schools was evident with the number of EU nationals’ children having increased from 150 to 2,500 in five years.
1.27. At a national level, 18% of homelessness acceptances in 2015-16 were foreign nationals – more than double the number in 2009-10 – with implications for who gets priority for social housing."